♦ù θn ❡st •✬❊▼❱ ❞✉ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉ ϑ✳ ❙♦✉s •❡s ❝♦♥❞✐t✐♦♥s ❞❡ ré❣✉•❛r✐tés✱ ♥♦✉s ♠❡♥t✐♦♥♥♦♥s q✉❡ •✬❊▼❱ ❡st ❝♦♥s✐st❛♥t✱ ❛s②♠♣t♦t✐q✉❡♠❡♥t ♥♦r♠❛•❡ ❡t ❛❞♠❡t ❡♥ ♣•✉s •❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ √ n θn -ϑ = 1 √ nI(ϑ) n j=1 ḟ (ϑ, X j ) f (ϑ, X j ) + o(1), I(ϑ) = ∞ -∞ ḟ (ϑ, x) 2 f (ϑ, x) ❞x. ◆♦t♦♥s q✉❡ I(ϑ) ❡t f (ϑ, x) ❞és✐❣♥❡♥t •✬✐♥❢♦r♠❛t✐♦♥ ❞❡ ❋✐s❤❡r ❡t •❛ ❞❡♥s✐té✱ r❡s♣❡❝✲ t✐✈❡♠❡♥t✳ ❈✐✲❞❡ss✉s ❡t ♣❛r •❛ s✉✐t❡ •❡ ♣♦✐♥t r❡♣rés❡♥t❡ •❛ ❞ér✐✈❛t✐♦♥ ♣❛r r❛♣♣♦rt à ϑ✳ ❊♥ ❛♣♣•✐q✉❛♥t •❛ ❢♦r♠✉•❡ ❞❡ ❚❛②•♦r✱ ♥♦✉s ♦❜t❡♥♦♥s ♣♦✉r •❛ st❛t✐st✐q✉❡ U n (•)
•

❊♥ ❡✛❡t✱ ♥♦✉s ❝♦♥s✐❞ér♦♥s •❡ ♠♦❞è•❡ ❞✬♦❜s❡r✈❛t✐♦♥s X n = (X 1 , . . . , X n ) ✐✳✐✳❞✳ ❡t ♥♦✉s ✉t✐•✐s♦♥s ♣❛r ❡①❡♠♣•❡ •✬❊❉▼ ♣♦✉r •❛ ❝♦♥str✉❝t✐♦♥ ❞✉ t❡st ❞❡ •✬❤②♣♦t❤ès❡ H 0 ❞♦♥♥é ♣❛r ✭✶✮✳ ◆♦✉s ✐♥tr♦❞✉✐s♦♥s •❛ st❛t✐st✐q✉❡ ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥✲▼✐s❡s φ * n = +∞ -∞ ξ n (x) 2 dF (ϑ * n , x) , ξ n (x) = √ n Fn (x) -F (ϑ * n , x) , ✭✹✮ ♦ù ϑ * n ❡st •✬❊❉▼ q✉✐ ❡st s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ❞❡ ❞✐st❛♥❝❡ ♠✐♥✐♠❛•❡ +∞ -∞ Fn (x) -F (ϑ * n , x) Ḟ (ϑ * n , x) dF (ϑ * n , x) = 0. ◆♦✉s ❞é✜♥✐ss♦♥s u * n = √ n (ϑ * n -ϑ)✳ ❆✐♥s✐✱ ϑ * n = u * n √ n +ϑ ❡t ❡♥ ✉t✐•✐s❛♥t •❛ ❢♦r♠✉•❡ ❞❡ ❚❛②•♦r •✬éq✉❛t✐♦♥ ♣ré❝é❞❡♥t❡ ❛❞♠❡t •❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ +∞ -∞ Fn (x) -F (ϑ, x) - u * n √ n Ḟ ( θ, x) Ḟ (ϑ * n , x) dF (ϑ * n , x) = 0. P❛r ❝♦♥séq✉❡♥t✱ •❛ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ u * n ❡st r❡♣rés❡♥té❡ ❝♦♠♠❡ s✉✐t ✿ u * n = +∞ -∞ Ḟ ( θ, x) Ḟ (ϑ * n , x)dF (ϑ * n , x) -1 +∞ -∞ B n (x) Ḟ (ϑ * n , x) dF (ϑ * n , x) , ✭✺✮ ♦ù B n (•) ❡st ❞é✜♥✐ ♣❛r ✭✷✮✳ P❛r •❡ t❤é♦rè♠❡ ❞❡ ❋✉❜✐♥✐✱ ♥♦✉s ❛✈♦♥s +∞ -∞ B n (x) Ḟ (ϑ * n , x) dF (ϑ * n , x) = √ n +∞ -∞ +∞ -∞ ✶ ■ {y<x} ❞ Fn (y) -f (ϑ, y) ❞y Ḟ (ϑ * n , x) dF (ϑ * n , x) ✽ ■◆❚|❖❉❯❈❚■❖◆ = √ n +∞ -∞ +∞ y Ḟ (ϑ * n , x) dF (ϑ * n , x) ❞ Fn (y) -f (ϑ, y) ❞y = √ n +∞ -∞ H (ϑ * n , y) ❞ Fn (y) -f (ϑ, y) ❞y , ❛✈❡❝ H (ϑ * n , y) = +∞ y Ḟ (ϑ * n , x) dF (ϑ * n , x)✳ ❊♥ ✉t✐•✐s❛♥t •✬❡①♣r❡ss✐♦♥ s✉✐✈❛♥t❡ ✿ +∞ -∞ H (ϑ * n , y) ❞ Fn (y) = 1 n n j=1 H (ϑ * n , X j ) ❡t •❡ t❤é♦rè♠❡ ❝❡♥tr❛• •✐♠✐t❡✱ ♥♦✉s ♦❜t❡♥♦♥s +∞ -∞ B n (x) Ḟ (ϑ * n , x) dF (ϑ * n , x) = 1 √ n n j=1 H (ϑ * n , X j ) - √ n +∞ -∞ H (ϑ * n , y) f (ϑ, y) ❞y = 1 √ n n j=1 (H (ϑ * n , X j ) -E ϑ H (ϑ * n , X j )) =⇒ N 0, σ(ϑ) 2 , ♦ù σ(ϑ) 2 = E ϑ (H (ϑ, X 1 ) -E ϑ H (ϑ, X 1 )) 2 . ❆✐♥s✐✱ •❛ ✈❛r✐❛❜•❡ ❛•é❛t♦✐r❡ u * n ❝♦♥✈❡r❣❡ ❡♥ •♦✐ ✈❡rs •❛ •♦✐ ♥♦r♠❛•❡ u * n =⇒ N 0, d(ϑ) 2 , d(ϑ) 2 = E ϑ Ḟ (ϑ, X 1 ) 2 -2 σ(ϑ) 2 . ❉❡ ♣•✉s✱ ❣râ❝❡ à •❛ ❝♦♥s✐st❛♥❝❡ ❞❡ •✬❊❉▼✱ •❡ ♣r♦❝❡ss✉s u * n ❞é✜♥✐ ♣❛r ✭✺✮ ❛❞♠❡t •❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ u * n = +∞ -∞ Ḟ (ϑ, x) 2 dF (ϑ, x) -1 +∞ -∞ H(ϑ, y) ❞B n (y) (1 + o(1)) , ♦ù B n (•) ❡st ❞é✜♥✐ ♣❛r ✭✷✮✳ ▼❛✐♥t❡♥❛♥t✱ ♥♦✉s ❛••♦♥s ét✉❞✐❡r •❡ ❝♦♠♣♦rt❡♠❡♥t ❛s②♠♣t♦t✐q✉❡ ❞❡ •❛ st❛t✐s✲ t✐q✉❡ ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥✲▼✐s❡s ❞é✜♥✐❡ ♣❛r ✭✹✮✳ ❊♥ ❡✛❡t✱ •❛ ❢♦r♠✉•❡ ❞❡ ❚❛②•♦r ❡t •❛ ❝♦♥✈❡r❣❡♥❝❡ B n (x) =⇒ B (F (ϑ, x))✱ ❛✈❡❝ B (F (ϑ, x)) ✉♥ ♣♦♥t ❇r♦✇♥✐❡♥✱ ♥♦✉s ❞♦♥♥❡♥t •❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ ξ n (x) = B n (x) - √ n (ϑ * n -ϑ) Ḟ ( θ, x) + o(1) =⇒ B (F (ϑ, x)) - +∞ -∞ Ḟ (ϑ, y) 2 dF (ϑ, y) -1 +∞ -∞
H (ϑ, y) ❞B (F (ϑ, y)) Ḟ (ϑ, x) .

■◆❚|❖❉❯❈❚■❖◆ ✾ ❊♥s✉✐t❡✱ s✐ ♥♦✉s ❝❤❛♥❣❡♦♥s •❡s ✈❛r✐❛❜•❡s t = F (ϑ, x) ❡t s = F (ϑ, y)✱ ♥♦✉s ♦❜t❡✲ g (ϑ, s) = (g 1 (ϑ, s) , g 2 (ϑ, s) , g 3 (ϑ, s)) ❡t h (ϑ, s) = (h 1 (ϑ, s) , h 2 (ϑ, s) , h 3 (ϑ, s))✳

♥♦♥s ξ n (x) =⇒ B (t) -J (ϑ) -1 1 0 H ϑ, F -1 ϑ (s) ❞B (s) t 0 ḟ ϑ, F -1 ϑ (s) f ϑ, F -1 ϑ (s) ❞s, ♦ù B(t), 0 ≤ t ≤ 1 ❡st ✉♥ ♣♦♥t ❇r♦✇♥✐❡♥✱ F -1 ϑ (s) ❡st •❛ ❢♦♥❝t✐♦♥ ✐♥✈❡rs❡ ❞❡ F (ϑ, y) ❡t J (ϑ) = 1 0 Ḟ ϑ, F -1 ϑ (s) 2 ds. ❆✐♥s✐✱ ξ n (x) =⇒ B (t) - 1 0 g (ϑ, s) ❞B (s) t 0 h (ϑ, s) ❞s, ❛✈❡❝ g (ϑ, s) = H ϑ, F -1 ϑ (s) 1 0 H ϑ, F -1 ϑ (v) 2 ❞v -1 2 , 1 0 g (ϑ, s) 2 ds = 1 ❡t h (ϑ, s) = J (ϑ) -1 ḟ ϑ, F -1 ϑ (s) f ϑ, F -1 ϑ (s) 1 0 H ϑ, F -1 ϑ (v) 2 ❞v 1 2 . ❋✐♥❛•❡♠❡♥t✱ s♦✉s •✬❤②♣♦t❤ès❡ H 0 ✱ ♥♦✉s ❛✈♦♥s ❝❡tt❡ ❝♦♥✈❡r❣❡♥❝❡ φ * n =⇒ φ * ≡ 1 0 U (t) 2 ❞t, ♦ù U (t) = B (t) - 1 0 g (ϑ, s) ❞B (s) t 0 h (ϑ, s) ❞s, 0 ≤ t ≤ 1. ✭✻✮ ▼❛✐♥t❡♥❛♥t✱ ♥♦✉s ♣♦✉✈♦♥s r❡♠♣•❛❝❡r •❡ ♣♦♥t ❇r♦✇♥✐❡♥ B(t), 0 ≤ t ≤ 1 ♣❛r B(t) = W (t) -W (1)t✱ ♦ù W (t) ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✳ ❆✐♥s✐✱ ♥♦✉s ♦❜t❡♥♦♥s U (t) = W (t) -W (
❆•♦rs✱ ✐• ❡st ❞✐✣❝✐•❡ ❞❡ tr♦✉✈❡r ✉♥❡ t❡••❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ♣♦✉r ❝❡ ♣r♦❝❡s✲ s✉s •✐♠✐t❡✱ ♠ê♠❡ ❞❛♥s •❡ ❝❛s ♦ù •❡ ♣❛r❛♠ètr❡ ϑ ❡st ✉♥✐❞✐♠❡♥s✐♦♥♥❡•✳ ❈❡❝✐ ❡st ♣r♦❜❛❜•❡♠❡♥t •❛ r❛✐s♦♥ ♣♦✉r •❛q✉❡••❡ ❝❡ ♣r♦❜•è♠❡ ♥✬❛ ♣❛s été ❝♦♥s✐❞éré •✉sq✉✬à ♠❛✐♥t❡♥❛♥t✳ ❆✐♥s✐✱ ❝❡tt❡ t❤ès❡ ❡st ❝♦♥s❛❝ré❡ ❛✉ ♣r♦❜•è♠❡ ❞❡ ❝♦♥str✉❝t✐♦♥ ❞✬✉♥❡ tr❛♥s✲ ❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❛♣♣•✐q✉é❡ à ✉♥ ♣r♦❝❡ss✉s U (•) ❛❞♠❡tt❛♥t •❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ U (t) = W (t) - 1 0 g (ϑ, s) ❞W (s) t 0 h (ϑ, s) ❞s, 0 ≤ t ≤ 1, ✭✼✮ ♦ù W (t), 0 ≤ t ≤ 1 ❡st ✉♥ ♣
1 0 g (ϑ, r) 2 dr = 1, ✭✽✮ ♦ù W (ν) , 0 ≤ ν ≤ 1 ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r ❡t h (•, •) ❡t g (•, •) s♦♥t ❞❡s ❢♦♥❝t✐♦♥s q✉✐ s❡r♦♥t ❞é✜♥✐❡s ❞❛♥s •❛ s✉✐t❡✳ ❆✐♥s✐✱ •❡ ♣r♦❜•è♠❡ s❡ ré❞✉✐t à ❛♣♣•✐q✉❡r •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L[•] ❛✉ ♣r♦❝❡ss✉s •✐♠✐t❡ U (•) t❡••❡ q✉❡ L [U ] (ν) = w ν ✱ ♦ù w ν , 0 ≤ ν ≤ 1 ❡st ✉♥ ♣
• Pr♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑✳ ◆♦✉s ❝♦♥s✐❞ér♦♥s •❡ ♣r♦❝❡s✲ s✉s X ε = (X t , 0 ≤ t ≤ T )✱ ❛✈❡❝ ❞X t = S (ϑ, X t ) ❞t + ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ♦ù W t ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✱ S (ϑ, x) ❡st ✉♥❡ ❢♦♥❝t✐♦♥ ré❣✉•✐èr❡ ❡t ❝♦♥♥✉❡ ❡t ε → 0✳ • Pr♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡✳ ◆♦✉s ❝♦♥s✐❞ér♦♥s X n = X t , 0 ≤ t ≤ T = nτ •❡ ♣r♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡ ❡t τ ✲♣ér✐♦❞✐q✉❡✱ ♦ù ✶✷ ■◆❚|❖❉❯❈❚■❖◆ τ > 0 ❡st ✉♥❡ ♣ér✐♦❞❡ ❝♦♥♥✉❡✱ n → ∞ ❡t ♦♥ ❛ E ϑ X t = Λ (ϑ, t) = t 0 λ (ϑ, s) ❞s.
▲❛ ❢♦♥❝t✐♦♥ ❞✬✐♥t❡♥s✐té λ (ϑ, s) ❡st ré❣✉•✐èr❡✱ ♣ér✐♦❞✐q✉❡ ❡t ❝♦♥♥✉❡✳ U (ν) 2 ❞ν, ♣r♦❝❡ss✉s u (t) , 0 ≤ t ≤ T ❞é✜♥✐ ♣❛r ✭✶✳✶✺✮✳ ❊♥s✉✐t❡✱ •❛ tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✶✻✮ ❞❡

• Pr♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❡r❣♦❞✐q✉❡✳ ◆♦✉s ❝♦♥s✐❞ér♦♥s •❡ ♣r♦❝❡ss✉s X T = (X t , 0 ≤ t ≤ T )✱ ❛✈❡❝ ❞X t = S (ϑ, X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ♦ù S (ϑ, x) ❡t σ (x) s♦♥t ❞❡✉① ❢♦♥❝t✐♦♥s ré❣✉•✐èr❡s ❡t ❝♦♥♥✉❡s ❡t T → ∞✳ ▲❡ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉ ϑ ∈ Θ = (
• Pr♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡ •♦rsq✉❡ T = nτ → ∞ ∆ n =⇒ 1 0 U (ν) 2 ❞ν, δ n =⇒ 1 0 U (ν) 2 ❞ν, • Pr♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❡r❣♦❞✐q✉❡ •♦rsq✉❡ T → ∞ ∆ * T =⇒ 1 0 U (ν) 2 ❞ν. ◆♦✉s r❛♣♣❡•♦♥s q✉❡ •❡s ❢♦♥❝t✐♦♥s h (•, •) ❡t g (•, •) s♦♥t
♣♣♦s♦♥s q✉❡ ♥♦✉s ♦❜s❡r✈♦♥s ✉♥ ♣r♦❝❡ss✉s X ε = (X t , 0 ≤ t ≤ T ) ❡♥ t❡♠♣s ❝♦♥t✐♥✉✱ s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡••❡ st♦❝❤❛st✐q✉❡ s✉✐✈❛♥t❡ ✿ ❞X t = S(X t ) ❞t + ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ✭✶✵✮ ♦ù W t , 0 ≤ t ≤ T ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✱ S(x) ❡st ✉♥❡ ❢♦♥❝t✐♦♥ ✐♥❝♦♥♥✉❡ ❡t ré❣✉•✐èr❡ ❡t •❛ ✈❛•❡✉r ✐♥✐t✐❛•❡ x 0 ❡st ❞ét❡r♠✐♥✐st❡✳ ◆♦✉s ❝♦♥str✉✐s♦♥s •❡ t❡st ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❞❡ •✬❤②♣♦t❤ès❡ ♥✉••❡ ♣❛r❛♠étr✐q✉❡ H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ♦ù ϑ ❡st ✉♥ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉ ❡t ✉♥✐❞✐♠❡♥s✐♦♥♥❡•✳ ❉❛♥s ❙❡❝t✐♦♥ ✶✳✷✱ ♥♦✉s ♣r♦✲ ♣♦s♦♥s •✬❊❉▼ ♣♦✉r ❝❡ ♣❛r❛♠ètr❡ ϑ * ε = arg min ϑ∈Θ X -x(ϑ) , ♦ù • ❡st •❛ ♥♦r♠❡ L 2 [0, T ] ❞é✜♥✐❡ ♣❛r X -x(ϑ) 2 = T 0 (X t -x t (ϑ)) 2 ❞t. P✉✐s✱ ♥♦✉s ét✉❞✐♦♥s •❛ st❛t✐st✐q✉❡ ❞❡ ❜❛s❡ ❞❛♥s ❙❡❝t✐♦♥ ✶✳✸ u ε (t) = X t -x t (ϑ * ε ) ε S (ϑ * ε , X t ) , 0 ≤ t ≤ T, ♦ù x t (ϑ) ❡st •❛ s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ✭✶✵✮ •♦rsq✉❡ ε = 0✳ ◆♦t♦♥
u (t) ✈❡rs U t T , 0 ≤ t ≤ T ♥♦✉s ❞♦♥♥❡ •❛ r❡♣rés❡♥t❛t✐♦♥ ✭✽✮✳ ◆♦✉s ❝♦♥str✉✐s♦♥s ❛•♦rs •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L [•] ❞❛♥s ❚❤é♦rè♠❡ ✶✳✹✳✶ ❡t ♥♦✉s ♣r♦✉✈♦♥s q✉❡ L [U ] (ν) = w ν , ♦ù w ν , 0 ≤ ν ≤ 1 ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✳ ❈❡•❛ ♥♦✉s ♣❡r♠❡t ❞❡ ❝♦♥str✉✐r❡ ✉♥ t❡st ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❞❛♥s ❚❤é♦rè♠❡ ✶✳✺✳✷✳ ❊♥ ❡✛❡t✱ ♥♦✉s ♦❜t❡♥♦♥s •❛ ❝♦♥✈❡r❣❡♥❝❡ s✉✐✈❛♥t❡ ✿ ∆ ε = 1 T T 0 L [U ε ] (t) 2 dt =⇒ 1 0 w 2 ν dν, ♦ù U ε (•) ❡st •❡ ♣r♦❝❡ss✉s ❞é✜♥✐ ♣❛r ✭✶✳✹✺✮ ❞❛♥s ❙❡❝t✐♦♥ ✶✳✺✳ P❛r ❝♦♥séq✉❡♥t✱ ♥♦✉s ❝♦♥str✉✐s♦♥s •❡ t❡st ❆❉❋ ψ ε = ✶ ■ {∆ε>cα} ✈✉ q✉❡ •❛ ❞✐str✐❜✉t✐♦♥ •✐♠✐t❡ ♥❡ ❞é♣❡♥❞ ♥✐ ❞❡ S (•) ♥✐ ❞❡ ϑ✳ ◆♦✉s ❞é✜♥✐ss♦♥s c α ❝♦♠♠❡ •❡ (1 -α)✲q✉❛♥t✐•❡ ❞❡ •❛ st❛t✐st✐q✉❡ •✐♠✐t❡✱ ❝✬❡st à ❞✐r❡✱ ✐• ❡st s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ s✉✐✈❛♥t❡ ✿ P 1 0 w 2 ν dν > c α = α, α ∈ (0, 1). ◆♦✉s ♠❡♥t✐♦♥♥♦♥s q✉❡ •❡ t❡st ψ ε ❡st ❞❡ ♥✐✈❡❛✉ ❛s②♠♣t♦t✐q✉❡ α✱ ❝✬❡st à ❞✐r❡✱ ♥♦✉s ❛✈♦♥s •♦rsq✉❡ ε → 0 E ϑ ψ ε = α + o(1), ♣♦✉r t♦✉s ϑ ∈ Θ, ♦ù E ϑ ❡st •✬❡s♣ér❛♥❝❡ ♠❛t❤é♠❛t✐q✉❡ s♦✉s •✬❤②♣♦t❤ès❡ ♥✉••❡ H 0 ✳ ❋✐♥❛•❡♠❡♥t✱ ❞❛♥s ❙❡❝t✐♦♥ ✶✳✻✱ ♥♦✉s tr❛✐t♦♥s •❡ ❝❛s ❞✬✉t✐•✐s❛t✐♦♥ ❞❡ •✬❊▼❱ ♣♦✉r •❡ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉✳ ❊♥ ❡✛❡t✱ •❛ r❡♣rés❡♥t❛t✐♦♥ ❞✉ ♣r♦❝❡ss✉s •✐♠✐t❡ ❡st ❝♦♠♠❡ s✉✐t ✿ U (ν) = W (ν) - 1 0 h (ϑ, s) dW (s) ν 0 h (ϑ, s) ds, 1 0 h (ϑ, s) 2 ds = 1 ❡t ♥♦tr❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✷✻✮ ❝♦ï♥❝✐❞❡ ❛✈❡❝ ❝❡••❡ ♣r♦♣♦sé❡ ❞❛♥s ❬✸✵(✳ ❉❛♥s ❈❤❛♣✐tr❡ ✷✱ ♥♦✉s s♦♠♠❡s ✐♥tér❡ssés ♣❛r •❛ ❝♦♥str✉❝t✐♦♥ ❞❡ ❞❡✉① t❡sts ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❜❛sés s✉r •❡s st❛t✐st✐q✉❡s ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ❡t ❞❡ ✏s❝♦r❡✲❢✉♥❝t✐♦♥✑✳ ❈❡tt❡ ❝♦♥str✉❝t✐♦♥ ❡st ✐♥tr♦❞✉✐t❡ ❡♥ ✉t✐•✐s❛♥t •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L [•] ❞é✜♥✐❡ ♣❛r ✭✶✳✷✻✮✳ ❊♥ ❡✛❡t✱ ♥♦✉s ♦❜s❡r✈♦♥s ✉♥ ♣r♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡ ♣ér✐♦❞✐q✉❡ X n = X t , 0 ≤ t ≤ T = nτ ❛✈❡❝ ✉♥❡ ♣ér✐♦❞❡ ❝♦♥♥✉❡ ■◆❚|❖❉❯❈❚■❖◆ ✶✺ τ > 0✳ ▲❛ ♠♦②❡♥♥❡ Λ (t) ❡t •❛ ❢♦♥❝t✐♦♥ ❞✬✐♥t❡♥s✐té λ (t) ✈ér✐✜❡♥t •❡s r❡•❛t✐♦♥s s✉✐✈❛♥t❡s ✿ E ϑ X t = Λ (ϑ, t) , Λ (ϑ, t) = t 0 λ (ϑ, s) ❞s s♦✉s •✬❤②♣♦t❤ès❡ ♥✉••❡ ♣❛r❛♠étr✐q✉❡ H 0 : Λ(•) ∈ {Λ(ϑ, •), ϑ ∈ Θ = (a, b)} , ♦ù a > 0 ❡t b < ∞✳ ◆♦✉s ✐♥tr♦❞✉✐s♦♥s ❞❛♥s ❙❡❝t✐♦♥ ✷✳✷ •✬❊❉▼ ϑ * n = arg min ϑ∈Θ Λn (•) -Λ (ϑ, •) , ♦ù Λn (•) ❡st •❛ ❢♦♥❝t✐♦♥ ❞❡ ♠♦②❡♥♥❡ ❡♠♣✐r✐q✉❡ ❞é✜♥✐❡ ♣❛r ✭✷✳✷✮ ❡t • ❡st •❛ ♥♦r♠❡ L 2 (0, τ )✱ ❝✬❡st à ❞✐r❡✱ Λn (•) -Λ (ϑ, •) 2 = τ 0 Λn (s) -Λ (ϑ, s) 2 ❞s. ❉❛♥s ❙❡❝t✐♦♥ ✷✳✸✱ ♥♦✉s ❝♦♥s✐❞ér♦♥s •❛ st❛t✐st✐q✉❡ ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ∆ n = n Λ (ϑ * n , τ ) 2 τ 0 Λn (r) -Λ (ϑ * n , r) 2 λ (ϑ * n , r) ❞r, ◆♦t♦♥s q✉❡ ❞❛♥s ❝❡tt❡ s❡❝t✐♦♥ ♥♦✉s ♦❜t❡♥♦♥s ✭s♦✉s •✬❤②♣♦t❤ès❡ H 0 ✮ •❛ ❝♦♥✈❡r❣❡♥❝❡ s✉✐✈❛♥t❡ ✿ ∆ n =⇒ 1 0 U (ν) 2 ❞ν, ♦ù •❡ ♣r♦❝❡ss✉s U (•) ❡st ❞é✜♥✐ ♣❛r ✭✽✮✳ ❊♥s✉✐t❡✱ ❞❛♥s ❙❡❝t✐♦♥ ✷✳✹✱ ♥♦✉s ❝♦♥str✉✐✲ s♦♥s ✉♥ ❛✉tr❡ t❡st ❆❉❋ ♣♦✉r •❡ ♠ê♠❡ ♠♦❞è•❡ ❞✬♦❜s❡r✈❛t✐♦♥s ❡♥ s❡ ❜❛s❛♥t s✉r •❛ st❛t✐st✐q✉❡ ❞❡ ✏s❝♦r❡✲❢✉♥❝t✐♦♥✑ u * n (t) = 1 √ n n j=1 t 0 λ(ϑ * n , v) λ(ϑ * n , v) [❞X j (v) -λ(ϑ * n , v) ❞v] . ◆♦✉s ♠❡♥t✐♦♥♥♦♥s q✉❡ ❝❡tt❡ st❛t✐st✐q✉❡ ❛ été ✐♥tr♦❞✉✐t❡ ♣❛r ❑✉t♦②❛♥ts ❬✹✸(✳ ❆✐♥s✐✱ ♥♦✉s ♠♦♥tr♦♥s •❛ ❝♦♥✈❡r❣❡♥❝❡ ❞❡ u * n (•) ✈❡rs •❡ ♣r♦❝❡ss✉s u (•) ❞é✜♥✐ ♣❛r ✭✷✳✸✽✮✳ P❛r ❛✐••❡✉rs✱ •❛ tr❛♥s❢♦r♠❛t✐♦♥ ✭✷✳✸✾✮ ♥♦✉s ❞♦♥♥❡ •❛ r❡♣rés❡♥t❛t✐♦♥ ✭✽✮ s❡•♦♥ ▲❡♠♠❡ ✷✳✹✳✺✳ ◆♦✉s r❡♠❛rq✉♦♥s q✉❡ ❞❛♥s •❡s ❞❡✉① ❝❛s✱ ❝✬❡st à ❞✐r❡✱ •❡s st❛t✐st✐q✉❡s ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ❡t ✏s❝♦r❡✲❢✉♥❝t✐♦♥✑✱ ♥♦✉s ♦❜t❡♥♦♥s •❡ ♠ê♠❡ ♣r♦❝❡ss✉s •✐♠✐t❡ ✭✽✮ ❛✈❡❝ ❞❡✉① ❢♦♥❝t✐♦♥s h (•, •) ❡t g (•, •) ❞✐✛ér❡♥t❡s✳ ◆♦✉s ♣❛ss♦♥s ❛•♦rs à ✶✻ ■◆❚|❖❉❯❈❚■❖◆ •✬❛♣♣•✐❝❛t✐♦♥ ❞❡ •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L[•] ❛✉ ♣r♦❝❡ss✉s U (•) ✭✈♦✐r ❚❤é♦rè♠❡ ✶✳✹✳✶✮ t❡• q✉❡ ✿ L[U ] (ν) = w ν , 0 ≤ ν ≤ 1. ▲❛ ♣rés❡♥t❛t✐♦♥ ❝✐✲❞❡ss✉s s✉❣❣èr❡ •❛ ❝♦♥str✉❝t✐♦♥ ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❡♥ ✉t✐•✐s❛♥t •❡s ✏✈❡rs✐♦♥s ❡♠♣✐r✐q✉❡s✑ U n (•, •) ❡t U * n (•) ❞❡ U (•) ✭❝❡s ❞❡✉① ♣r♦❝❡ss✉s s♦♥t ❞é✜♥✐s r❡s♣❡❝t✐✈❡♠❡♥t ♣❛r ✭✷✳✶✷✮ ❡t ✭✷✳✹✸✮✮ ❝♦♠♠❡ s✉✐t ✿ ◆♦✉s ✐♥tr♦❞✉✐s♦♥s •❡s st❛t✐st✐q✉❡s ❝♦rr❡s♣♦♥❞❛♥t❡s ❛✉① st❛t✐st✐q✉❡s ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ❡t ✏s❝♦r❡✲❢✉♥❝t✐♦♥✑ ∆ * n = τ 0 L[U n ](r) 2 Λ (ϑ * n , τ ) λ (ϑ * n , r) ❞r, δn = 1 0 L[U * n ](ν) 2 ❞ν. ❊♥s✉✐t❡✱ ♥♦✉s ♠♦♥tr♦♥s •❛ ❝♦♥✈❡r❣❡♥❝❡ ♣♦✉r ❝❡s ❞❡✉① st❛t✐st✐q✉❡s ✭s♦✉s •✬❤②♣♦✲ t❤ès❡ H 0 ✮ ∆ * n =⇒ 1 0 w 2 ν ❞ν, δn =⇒ 1 0 w 2 ν ❞ν ❡t ♣❛r ❝♦♥séq✉❡♥t✱ •❡s t❡sts ψ * n = ✶ ■ { ∆ * n >cα} , φn = ✶ ■ { δn>cα} , P 1 0 w 2 ν ❞ν > c α = α s♦♥t ❆❉❋ ❡t ❞❡ ♥✐✈❡❛✉ ❛s②♠♣t♦t✐q✉❡ α ∈ (0, 1)✳ ▲❡ ❈❤❛♣✐tr❡ ✸ ❡st ❝♦♥s❛❝ré à •❛ ❝♦♥str✉❝t✐♦♥ ❞✬✉♥ t❡st ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❞❛♥s •❡ ❝❛s ❞✉ ♣r♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❡r❣♦❞✐q✉❡ X T = {X t , 0 ≤ t ≤ T } ♦❜s❡r✈é ❡♥ t❡♠♣s ❝♦♥t✐♥✉ ❡t s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ❞✐✛ér❡♥t✐❡••❡ st♦❝❤❛st✐q✉❡ s✉✐✈❛♥t❡ ✿ ❞X t = S(X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T. ❙♦✉•✐❣♥♦♥s q✉❡ W t , 0 ≤ t ≤ T ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✱ S (x) ❡st ✉♥❡ ❢♦♥❝t✐♦♥ ✐♥❝♦♥♥✉❡✱ •❡ ❝♦❡✣❝✐❡♥t ❞❡ ❞✐✛✉s✐♦♥ σ (x) 2 > 0 ❡st ✉♥❡ ❢♦♥❝t✐♦♥ ❝♦♥♥✉❡ ❡t X 0 ❡st •❛ ✈❛•❡✉r ✐♥✐t✐❛•❡ ❞❡ X t ✳ ❙✉♣♣♦s♦♥s q✉❡ •✬❤②♣♦t❤ès❡ ❞❡ ❜❛s❡ ❡st ♣❛r❛♠étr✐q✉❡ H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ❝✬❡st à ❞✐r❡✱ •❡ ♣r♦❝❡ss✉s X T ❡st s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ s✉✐✈❛♥t❡ ✿ ❞X t = S(ϑ, X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ♦ù S (ϑ, x) ❡st ✉♥❡ ❢♦♥❝t✐♦♥ ré❣✉•✐èr❡✱ ❝♦♥♥✉❡ ❡t ❞é♣❡♥❞❛♥t❡ ❞❡ ϑ✳ ■◆❚|❖❉❯❈❚■❖◆ ✶✼ ◆♦✉s ♣r♦♣♦s♦♥s •✬❊❉▼ ϑ * T ♣♦✉r •❡ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉ ϑ ❞❛♥s ❙❡❝t✐♦♥ ✸✳✸ ✿ ϑ * T = arg inf ϑ∈Θ R fT (x) -f (ϑ, x) 2 ❞x. ◆♦t♦♥s q✉❡ fT (x) ❡st •✬❡st✐♠❛t❡✉r ❞❡ t❡♠♣s •♦❝❛• ❞❡ •❛ ❞❡♥s✐té ✐♥✈❛r✐❛♥t❡ f (ϑ, x) ✭❞é✜♥✐s ♣❛r ✭✸✳✾✮ ❡t ✭✸✳✽✮✱ r❡s♣❡❝t✐✈❡♠❡♥t✮✳ ❉❛♥s ❙❡❝t✐♦♥ ✸✳✹✱ ♥♦✉s ❝♦♥s✐❞ér♦♥s •❛ st❛t✐st✐q✉❡ ❞❡ t②♣❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s δ T = ∞ -∞ η * T (ϑ * T , x) 2 ❞F (ϑ * T , x) , ❛✈❡❝ η * T (ϑ * T , x) = √ T fT (x) -f (ϑ * T , x) =⇒ η (ϑ, x) = 2 f (ϑ, x) +∞ -∞ F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) ❞W (y) -2 J (ϑ) -1 R R F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) f (ϑ, x) ḟ (ϑ, x) ❞x ❞W (y) ḟ (ϑ, x). ❯♥❡ ♣r❡♠✐èr❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❛ été ❛♣♣•✐q✉é❡ ✭✈♦✐r ❑✉t♦②❛♥ts ❬✹✶(✮ L 1 [η] (x) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) = U (F (ϑ, x)) ♣♦✉r •❡ ♣r♦❝❡ss✉s •✐♠✐t❡ η (•, •)✳ ❆✐♥s✐✱ ♥♦✉s ♦❜t❡♥♦♥s ❞❛♥s ▲❡♠♠❡ ✸✳✹✳✻ •❛ r❡♣ré✲ s❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ U (t) = W (t) - 1 0 g (ϑ, s) ❞W (s) t 0 h (ϑ, s) ❞s, 0 ≤ t ≤ 1, ♦ù W (t) , 0 ≤ t ≤ 1 ❡st ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✳ ▲✬ét❛♣❡ s✉✐✈❛♥t❡ s✬❛❣✐t ❞✬✐♥✲ tr♦❞✉✐r❡ ✉♥❡ s❡❝♦♥❞❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L [•] ✭✶✳✷✻✮ ❝♦♠♠❡ ❞❛♥s •❡s s❡❝t✐♦♥s ♣ré❝é❞❡♥t❡s ❡t ♦❜t❡♥✐r •✬é❣❛•✐té s✉✐✈❛♥t❡ ✿ L [U ] (t) = w t , 0 ≤ t ≤ 1. P❛r ❝♦♥séq✉❡♥t✱ ♥♦✉s ❛✈♦♥s •❛ ❝♦♥✈❡r❣❡♥❝❡ δ * T = ∞ -∞ L [ξ T ] (x) 2 dF (ϑ * T , x) =⇒ 1 0 w 2 t dt, ♦ù ξ T (•, •) ❡st •❡ ♣r♦❝❡ss✉s ❞é✜♥✐ ♣❛r ✭✸✳✷✷✮ q✉✐ ❡st ❛s②♠♣t♦t✐q✉❡♠❡♥t éq✉✐✈❛•❡♥t à L 1 [η * T ] (•)✳ ❉♦♥❝✱ •❡ t❡st ψ * T = ✶■ {δ * T >cα} ❡st ❞❡ ♥✐✈❡❛✉ ❛s②♠♣t♦t✐q✉❡ α ∈ (0, 1) ❈❤❛♣t❡r ✶ ❖♥ •♦❋ ❚❡st ❢♦r P❡rt✉r❜❡❞ ❉②♥❛♠✐❝❛• ❙②st❡♠s ✶✳✶ ■♥tr♦❞✉❝t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜•❡♠ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❛s②♠♣t♦t✐❝❛••② ❞✐str✐❜✉t✐♦♥ ❢r❡❡ t❡st ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❛ ✏s♠❛•• ♥♦✐s❡✑ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✳ ❯♥❞❡r t❤❡ ♥✉•• ❤②♣♦t❤❡s✐s✱ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t ❞❡✲ ♣❡♥❞s ♦♥ s♦♠❡ ✉♥❦♥♦✇♥ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ♣❛r❛♠❡t❡r✳ ❚❤❡r❡❢♦r❡ t❤❡ ❜❛s✐❝ ✭♥✉••✮ ❤②♣♦t❤❡s✐s ✐s ♣❛r❛♠❡tr✐❝✳ •♦♦❞♥❡ss✲♦❢✲✜t t❡sts ♣•❛② ❛♥ ✐♠♣♦rt❛♥t r♦•❡ ✐♥ t❤❡♦r❡t✐❝❛• ❛♥❞ ❛♣♣•✐❡❞ st❛t✐s✲ t✐❝s✳ ❚❤❡② ❛••♦✇ t♦ ✈❡r✐❢② t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣r♦♣♦s❡❞ t❤❡♦r❡t✐❝❛• ♠♦❞❡•s ❛♥❞ r❡❛• ❞❛t❛✳ ■♥ ❢❛❝t✱ t❤❡s❡ t❡sts ❛r❡ ✇✐❞❡•② ✉s❡❞ ❛s ♠❛t❤❡♠❛t✐❝ ♠♦✲ ❞❡•s ✐♥ ♠❛♥② ✜❡•❞s ❛♥❞ st✉❞✐❡❞ ✈❡r② ✐♥t❡♥s✐✈❡•② ✐♥ st❛t✐st✐❝s s✐♥❝❡ ✶✾✷✽ ❞✉❡ t♦ t❤❡ ✇♦r❦s ❜② ❈r❛♠ér ✭✶✾✷✽✮✱ ✈♦♥ ▼✐s❡s ✭✶✾✸✶✮✱ ❑♦•♠♦❣♦r♦✈ ✭✶✾✸✸✮ ❛♥❞ ❙♠✐r♥♦✈ ✭✶✾✸✻✮ ✭s❡❡✱ ❡✳❣✳✱ ❬✶✸( ❛♥❞ ❬✶✽(✮✳ ▲❡t ✉s r❡❝❛•• t❤❡ ✇❡••✲❦♥♦✇♥ ❜❛s✐❝ r❡s✉•ts ❢♦r t❤❡ ✐✳✐✳❞✳ ♠♦❞❡•✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♥✉•• ❤②♣♦t❤❡s✐s ✐s s✐♠♣•❡✳ ❉❡♥♦t✐♥❣ t❤❡ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✉♥❞❡r t❤❡ ♥✉•• ❤②♣♦t❤❡s✐s ❜② F 0 (x)✱ ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ ✐❢ t❤❡ ✐✳✐✳❞✳ ♦❜s❡r✈❛t✐♦♥s X n = (X 1 , . . . , X n ) ❤❛✈❡ t❤✐s ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥✳ ▼❛♥② •♦❋ t❡sts ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt② ✿ ✉♥❞❡r t❤❡ ♥✉•• ❤②♣♦t❤❡s✐s✱ t❤❡ ♥♦r♠❛•✐③❡❞ ❡♠♣✐r✐❝❛• ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ √ n Fn (x) -F 0 (x) , Fn (x) = 1 n n j=1 ✶ ■ {X j <x} ✶✾ ✷✵ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ t❤❡ ❇r♦✇♥✐❛♥ ❜r✐❞❣❡ B (F 0 (x))✳ ■♥ ♣❛rt✐❝✉•❛r✱ ❢♦r t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s st❛t✐st✐❝ ✇❡ ❤❛✈❡ ✭✇✐t❤ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ s = F 0 (x)✮ φ n = n ∞ -∞ Fn (x) -F 0 (x) 2 dF 0 (x) =⇒ 1 0 B (s) 2 ds ≡ φ. ❚❤❡ •✐♠✐t φ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ F 0 (•) ✭❞✐str✐❜✉t✐♦♥ ❢r❡❡✮✳ ❚❤❡r❡❢♦r❡ t❤❡ ❈r❛♠ér✲ ✈♦♥ ▼✐s❡s t❡st Ψ n = ✶■ {φn>cα} , P {φ > c α } = α ✐s ❆❉❋ ❛♥❞ ❤❛s t❤❡ ❛s②♠♣t♦t✐❝ (n → ∞) s✐③❡ α ∈ (0, 1) ✭s❡❡✱ ❡✳❣✳✱ ❬✹✻(✮✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 : F 0 (•) ∈ {F 0 (ϑ, •) , ϑ ∈ Θ = (a, b)}, ✇❤❡r❡ ϑ ✐s s♦♠❡ ✉♥❦♥♦✇♥ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ♣❛r❛♠❡t❡r✱ t❤❡ s✐♠✐•❛r •✐♠✐t U n (x) = √ n Fn (x) -F 0 ( θn , x) =⇒ B (F 0 (ϑ, x)) -ζ Ḟ0 (ϑ, x) ≡ u (x) ✐s ♥♦ ♠♦r❡ ❞✐str✐❜✉t✐♦♥ ❢r❡❡ ❛♥❞ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ t❤r❡s❤♦•❞ c α ✐s ♠✉❝❤ ♠♦r❡ ❝♦♠✲ ♣•✐❝❛t❡❞✳ ❍❡r❡ θn ✐s t❤❡ ♠❛①✐♠✉♠ •✐❦❡•✐❤♦♦❞ ❡st✐♠❛t♦r ✭▼▲❊✮ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✱ Ḟ0 (ϑ, •) ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ F 0 (ϑ, •) ✇✳r✳t✳ ϑ ❛♥❞ ζ ✐s ❛ •❛✉ss✐❛♥ ✈❛r✐❛❜•❡✳ ❇② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s t = F 0 (ϑ, x) ❛♥❞ s = F 0 (ϑ, y)✱ ✇❡ ❤❛✈❡ ❢♦r u (•) t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭s❡❡✱ ❡✳❣✳✱ ❉❛r•✐♥❣ ❬✶✸(✮ U (t) = B (t) - 1 0 h (ϑ, s) dB (s) t 0 h (ϑ, s) ds. ✭✶✳✶✮ ◆♦t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ h (•, •) ❛♣♣❡❛rs ❤❡r❡ t✇✐❝❡ ❜❡❝❛✉s❡ ✇❡ ✉s❡❞ t❤❡ ▼▲❊✳ ❋♦r t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r ✭▼❉❊✮ ✉s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ▼▲❊ ✇❡ ♦❜t❛✐♥ t❤❡ •✐♠✐t U (t) = B (t) - 1 0 g (ϑ, s) dB (s) t 0 h (ϑ, s) ds, 1 0 g (ϑ, s) 2 ds = 1 ✭✶✳✷✮ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s✳ ■♥ ❜♦t❤ ❝❛s❡s✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡sts ❛r❡ ♥♦ ♠♦r❡ ❆❉❋ ❛♥❞ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ t❤r❡s❤♦•❞ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ P 1 0 U (s) 2 ds > c α = α ❝❛♥ ❜❡ ❛ ❞✐✣❝✉•t ♣r♦❜•❡♠✳ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✷✶ ❚❤❡r❡ ❛r❡ s❡✈❡r❛• ♣♦ss✐❜✐•✐t✐❡s t♦ s♦•✈❡ t❤✐s ♣r♦❜•❡♠ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ •✐♠✐t ✭✶✳✶✮✳ ❖♥❡ ♦❢ t❤❡♠ ✐s t♦ ✜♥❞ ❛ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•] ♦❢ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ U (•)✱ s✉❝❤ t❤❛t L [U ] (t) = W (t)✱ ✇❤❡r❡ W (t) , 0 ≤ t ≤ 1 ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡♥ t❤❡ st❛t✐st✐❝ υn = ∞ -∞ L [U n ] (x) 2 dF 0 ( θn , x) =⇒ 1 0 W (t) 2 dt ≡ υ ❛♥❞ ✇❡ ♦❜t❛✐♥ ♦♥❝❡ ♠♦r❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢r❡❡ •✐♠✐t υ✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st Φn = ✶ ■ {υn>dα} , P {υ > d α } = α ✐s ❆❉❋✳ ❙✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥ ✇❛s ♣r♦♣♦s❡❞ ❜② ❑❤♠❛•❛❞③❡ ❬✸✵(✳ ❲❡ ❤❛✈❡ t♦ ❡♠✲ ♣❤❛s✐③❡ t❤❛t ✐♥ ❬✸✵( ❛♥❞ ✐♥ ♠❛♥② ♦t❤❡r ✇♦r❦s ✭s❡❡✱ ❡✳❣✳✱ t❤❡ ♣❛♣❡r ▼❛❣•❛♣❤❡r✐❞③❡ ❡t ❛•✳ ❬✹✾(✮ t❤❡ ❡st✐♠❛t♦r ✉s❡❞ ✇❛s ❛•✇❛②s t❤❡ ▼▲❊ ❛♥❞ t❤✐s ✐s ✐♠♣♦rt❛♥t ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤✐s •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ▼❛♥② ❛✉t❤♦rs ✇r♦t❡ t❤❛t s✐♠✐•❛r tr❛♥s❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦t❤❡r ❡st✐♠❛t♦rs ✇✐t❤ •✐♠✐t ✭✶✳✷✮✱ ❜✉t ❛s ✇❡ ❦♥♦✇ t❤✐s ✇♦r❦ ✭❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ♦t❤❡r ❡st✐♠❛t♦rs✮ ✇❛s ♥♦t ❞♦♥❡✳ ❚❤❡ ♣r♦❜•❡♠ ♦❢ •♦❋ t❡st✐♥❣ ❢♦r t❤❡ ♠♦❞❡• ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❛t✐♦♥s ♦❢ ✏s♠❛•• ♥♦✐s❡✑ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✱ ✇✐t❤ ❛ s✐♠♣•❡ ♥✉•• ❤②♣♦t❤❡s✐s Θ = {ϑ 0 }✱ ✇❛s st✉❞✐❡❞ ✐♥ ❬✶✷( ❛♥❞ ❬✹✵(✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♦❜s❡r✈❡❞ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✉♥❞❡r ❤②✲ ♣♦t❤❡s✐s ✐s ❞X t = S 0 (X t ) ❞t + εσ (X t ) ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ✭✶✳✸✮ ✇✐t❤ ❞❡t❡r♠✐♥✐st✐❝ ✐♥✐t✐❛• ✈❛•✉❡ x 0 ✱ ❦♥♦✇♥ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ε 2 σ (•) 2 > 0✱ ❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥ S 0 (•) ❛♥❞ W t , 0 ≤ t ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ◆♦t❡ t❤❛t ✐❢ t❤❡ ❢✉♥❝t✐♦♥s S 0 (x) ❛♥❞ σ (x) ❛r❡ ▲✐♣s❝❤✐t③ ✇✳r✳t✳ x✱ t❤❡♥ ✇❡ ❤❛✈❡ ✇✐t❤ ♣r♦❜❛❜✐•✐t② ✶ sup 0≤t≤T |X t -x t | ≤ Cε sup 0≤t≤T |W t | , ✇❤❡r❡ C ✐s s♦♠❡ ❝♦♥st❛♥t✳ ❚❤❡r❡❢♦r❡ t❤❡ ♣r♦❝❡ss X t ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠•② ✇✳r✳t✳ t ∈ [0, T ] ✇✐t❤ ♣r♦❜❛❜✐•✐t② ✶ t♦ x t = x t (ϑ 0 ) ✭s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✸✮ ❛s ε = 0✮✳ ❲❡ ❤❛✈❡ ❛s ✇❡•• sup 0≤t≤T E |X t -x t | 2 ≤ Cε 2 . ✷✷ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❋♦r t❤❡ ♣r♦♦❢ s❡❡ ❬✸✼(✱ ▲❡♠♠❛ ✶✳✶✸✳ ❚❤❡ •♦❋ t❡st ✇❛s ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ♥♦r♠❛•✐③❡❞ ❞✐✛❡r❡♥❝❡ ε -1 (X t -x t ) ❛♥❞ t❤❡ •✐♠✐t ♦❢ t❤✐s st❛t✐st✐❝ ✐s ❛ •❛✉ss✐❛♥ ♣r♦❝❡ss✳ ❚❤❡ •❛tt❡r ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❛ ❲✐❡♥❡r ♣r♦❝❡ss w s , 0 ≤ s ≤ 1 ❛s ❢♦••♦✇s ✿ ■♥tr♦❞✉❝❡ t❤❡ st❛t✐st✐❝ δ ε = T 0 σ (x t ) S 0 (x t ) 2 ❞t -2 T 0 X t -x t εS 0 (x t ) 2 2 σ (x t ) 2 ❞t. ❚❤❡ ❢♦••♦✇✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ δ ε =⇒ ∆ ≡ 1 0 w 2 s ❞s ✇❛s ♣r♦✈❡❞ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ t❡st φ ε = ✶ ■ {δε>cα} ✇✐t❤ P ∆ > c α = α ✐s ❆❉❋✳ ❚❤❡ ❝❛s❡ ♦❢ t❤❡ ❆❉❋ •♦❋ t❡sts ❢♦r ✏s♠❛•• ♥♦✐s❡✑ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s ✇✐t❤ ♣❛r❛♠❡tr✐❝ ❜❛s✐❝ ❤②♣♦t❤❡s✐s ✇❛s st✉❞✐❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ✐♥ ❬✷✻(✱ ❬✺✻( ❛♥❞ ❬✹✷(-❬✹✸(✳ ❋♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ t❡sts t❤❡ ❡st✐♠❛t♦r ✉s❡❞ ✇❛s ❛•✇❛②s t❤❡ ▼▲❊ ✭s❡❡ ❑✉t♦②❛♥ts ❬✸✻(✮✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❛r❡ s❡✈❡r❛• •♦❋ t❡sts ❢♦r t❤❡ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s ♣r♦♣♦s❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ✐♥ t❤❡ ✇♦r❦s ❬✺✶(✱ ❬✺✹( ❛♥❞ ❬✸✹(✳ ◆♦t❡ t❤❛t ✐♥ s♦♠❡ ♣r♦❜•❡♠s ✐t ✐s ♥♦t ♣♦ss✐❜•❡ t♦ ❤❛✈❡ ❛♥ ❡①♣•✐❝✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ▼▲❊ ❛♥❞ t❤❡r❡❢♦r❡ s♦♠❡t✐♠❡s ✐t ✐s ❜❡tt❡r t♦ ✉s❡ ♦t❤❡r ❡st✐♠❛t♦rs✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❡❛s✐•② ❝❛•❝✉•❛t❡❞✳ ❋♦r ❡①❛♠♣•❡✱ t❤✐s ❝❛♥ ❜❡ t❤❡ ▼❉❊✱ ❛ ♠❡t❤♦❞ ♦❢ ♠♦♠❡♥ts ❡st✐♠❛t♦r ♦r ❛ tr❛•❡❝t♦r② ✜tt✐♥❣ ❡st✐♠❛t♦r ❛♥❞ s♦ ♦♥✳ ■♥ ❛•• s✉❝❤ ❝❛s❡s✱ t❤❡ •✐♠✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ✉♥❞❡r•②✐♥❣ st❛t✐st✐❝s ✇✐•• ❜❡ •✐❦❡ ✭✶✳✷✮ ❜✉t ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s h (•, •) ❛♥❞ g (•, •) ✭s❡❡ ❜❡•♦✇✮✳ ■♥ t❤✐s ❈❤❛♣t❡r✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ❜② t✇♦ ♣r♦❜•❡♠s✳ ❲❡ ✜rst ❞❡❛• ✇✐t❤ t❤❡ ♣r♦❜•❡♠ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ❆•• st❡♣s ♦❢ t❤✐s ❝♦♥str✉❝t✐♦♥ ❛r❡ ❞❡✈❡•♦♣❡❞ ✐♥ ❙❡❝t✐♦♥ ✶✳✹✳ ❚❤❡♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ s❡❝♦♥❞ ♣r♦✲ ❜•❡♠ ✐♥ ❙❡❝t✐♦♥ ✶✳✺ ✿ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❆❉❋ •♦❋ t❡st ❜❛s❡❞ ♦♥ t❤✐s •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✇✐t❤ ✏s♠❛•• ♥♦✐s❡✑ ♦❜s❡r✈❡❞ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❚❤❡ ♠❛✐♥ r❡s✉•ts ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❚❤❡♦r❡♠ ✶✳✹✳✶ ❛♥❞ ❚❤❡♦r❡♠ ✶✳✺✳✷✱ r❡s♣❡❝t✐✈❡•②✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✶✳✹✳✶ ❛♥❞ t❤❡ ❆❉❋ t❡st ✐s ❣✐✈❡♥ ❜❡•♦✇ ✐♥ ❚❤❡♦r❡♠ ✶✳✺✳✷✳ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✷✸ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♦❜s❡r✈❡ ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♣r♦❝❡ss X ε = (X t , 0 ≤ t ≤ T )✱ ✇❤✐❝❤ ✐s t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❞X t = S(X t ) ❞t + ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ✭✶✳✹✮ ✇❤❡r❡ W t , 0 ≤ t ≤ T ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✱ S(x) ✐s s♦♠❡ ✉♥❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ❛♥❞ x 0 ✐s t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ✐♥✐t✐❛• ✈❛•✉❡✳ ❲❡ ❝♦♥str✉❝t ❛♥ ❆❉❋ •♦❋ t❡st ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ✇❤❡r❡ a > 0✱ b < ∞ ❛♥❞ ϑ ✐s ❛♥ ✉♥❦♥♦✇♥ ❛♥❞ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ♣❛r❛♠❡t❡r ❛♥❞ ✇❡ ♣r♦♣♦s❡ t❤❡ ▼❉❊ ❢♦r t❤✐s ♣❛r❛♠❡t❡r ϑ * ε = arg min ϑ∈Θ X -x(ϑ) , ✇❤❡r❡ • ✐s L 2 [0, T ] ♥♦r♠ ❞❡✜♥❡❞ ❜② X -x(ϑ) 2 = T 0 (X t -x t (ϑ)) 2 ❞t. ❍❡r❡ x t = x t (ϑ) ✐s t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✹✮ ❛s ε = 0✳ ❲❡ r❡❛•✐③❡ t❤❡ ❢♦••♦✇✐♥❣ ♣r♦❣r❛♠✳ ❋✐rst✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ❜❛s✐❝ st❛t✐st✐❝ u ε (t) = X t -x t (ϑ * ε ) ε S (ϑ * ε , X t ) , 0 ≤ t ≤ T ❝♦♥✈❡r❣❡s t♦ t❤❡ r❛♥❞♦♠ ♣r♦❝❡ss u (t) , 0 ≤ t ≤ T ✭s❡❡ ✭✶✳✶✺✮ ❜❡•♦✇✮✳ ◆♦t❡ t❤❛t t❤✐s st❛t✐st✐❝ ✇❛s ♣r♦♣♦s❡❞ ❜② ❑✉t♦②❛♥ts ❬✹✷( ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ▼▲❊✳ ❚❤❡♥ ✇❡ tr❛♥s❢♦r♠ u (t) ✐♥ U t T , 0 ≤ t ≤ T ✭s❡❡ ✭✶✳✶✻✮ ❜❡•♦✇✮✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♦❜t❛✐♥ ❢♦r U (•) t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U (ν) = W (ν) - 1 0 g (ϑ, r) dW (r) ν 0 h (ϑ, r) dr, 1 0 g (ϑ, r) 2 dr = 1. ✭✶✳✺✮ ❲❡ ✇✐•• ❣✐✈❡ •❛t❡r t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s g (•, •) ✭✶✳✷✵✮ ❛♥❞ h (•, •) ✭✶✳✶✾✮✳ ❚❤❡ •❛st st❡♣ ✐s t♦ ❛♣♣•② t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•] s✉❝❤ t❤❛t L [U ] (ν) = w ν , 0 ≤ ν ≤ 1✱ ✇❤❡r❡ w ν ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✭s❡❡ ❚❤❡♦r❡♠ ✶✳✹✳✶✮✳ ❚❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ ❢♦r♠ ♦❢ t❤✐s •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ✷✹ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❚❤❡♥ ✇❡ r❡❛•✐③❡ s✐♠✐•❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✹✹✮ ✇✐t❤ t❤❡ ✏❡♠♣✐r✐❝❛•✑ ♣r♦❝❡ss u ε (•)✱ ❛♣♣•② t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•] t♦ U ε (•) ❣✐✈❡♥ ❜② ✭✶✳✹✺✮ ❛♥❞ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✉♥❞❡r H 0 ✮ L [U ε ] -→ L [U ] (ν) = w ν , 0 ≤ ν ≤ 1. ▼♦r❡♦✈❡r✱ ✇❡ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∆ ε = 1 T T 0 L [U ε ] (t) 2 dt =⇒ 1 0 w 2 ν dν. ❚❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ ∆ ε ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ S (•, •) ❛♥❞ ϑ✳ ❚❤✐s ❛••♦✇s ✉s t♦ ❝♦♥str✉❝t t❤❡ ❆❉❋ t❡st ψ ε = ✶ ■ {∆ε>cα} ✐♥ ❚❤❡♦r❡♠ ✶✳✺✳✷✳ ❚❤❡ t❤r❡s❤♦•❞ c α ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ P 1 0 w 2 ν dν > c α = α, α ∈ (0, 1). ◆♦t❡ t❤❛t t❤❡ t❡st ψ ε ✐s ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α✱ ✐✳❡✳✱ ✇❡ ❤❛✈❡ ❛s ε → 0 E ϑ ψ ε = α + o(1), ❢♦r ❛•• ϑ ∈ Θ, ✇❤❡r❡ E ϑ ✐s t❤❡ ♠❛t❤❡♠❛t✐❝❛• ❡①♣❡❝t❛t✐♦♥ ✉♥❞❡r t❤❡ ♣❛r❛♠❡tr✐❝ ❤②♣♦t❤❡s✐s H 0 ✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✐❢ ✐♥ ♦✉r ♣r♦❜•❡♠ ✇❡ ✉s❡ t❤❡ ▼▲❊ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r✱ t❤❡♥ t❤❡ •✐♠✐t r❡♣r❡s❡♥t❛t✐♦♥ ✐s U (ν) = W (ν) - 1 0 h (ϑ, s) dW (s) ν 0 h (ϑ, s) ds, 1 0 h (ϑ, s) 2 ds = 1 ✭✶✳✻✮ ❛♥❞ ♦✉r tr❛♥s❢♦r♠❛t✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦♥❡ ♣r♦♣♦s❡❞ ✐♥ ❬✸✵(✳ ❲❡ ❞✐s❝✉ss t❤✐s ❝❛s❡ ✐♥ t❤❡ ❙❡❝t✐♦♥ ✶✳✻✳ ❲❡ ❤❛✈❡ t♦ ♥♦t❡ t❤❛t t❤✐s •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐s r❛t❤❡r ❝✉♠❜❡rs♦♠❡ ❛♥❞ t❤❡ r❡❛•✐③❛t✐♦♥ ♦❢ t❤❡ t❡st ❝❛♥ ❜❡ ❛ ❝♦♠♣✉t❛t✐♦♥❛••② ❞✐✣❝✉•t ♣r♦❜•❡♠ t♦♦✳ ❲❡ s✉♣♣♦s❡ t❤❛t t❤❡ ♣r❡s❡♥t❡❞ r❡s✉•t ✐s ♦❢ t❤❡♦r❡t✐❝❛• ✐♥t❡r❡st ❛♥❞ ❛••♦✇s t♦ ✏❝•♦s❡ t❤❡ ❣❛♣✑ ✐♥ t❤✐s ✜❡•❞✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ ✇❡ ✉♥❞❡rst❛♥❞ t❤❛t t❤✐s r❡s✉•t ✐s ✐♥ s♦♠❡ s❡♥s❡ ✏♥❡❣❛t✐✈❡✑ ❛♥❞ s❛②s t❤❛t ✐❢ ✇❡ ❤❛✈❡ ♥♦ ▼▲❊ ✐t ✐s ❜❡tt❡r t♦ s❡❡❦ ❛♥♦t❤❡r •♦❋ t❡st✱ ✇❤✐❝❤ ✐s ❆❉❋✳ ◆♦t❡ ❛s ✇❡•• t❤❛t ✐♥ ✐✳✐✳❞✳ ❝❛s❡✱ ❡✈❡♥ ✐❢ t❤❡ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡r ✐s ♦♥❡✲ ❞✐♠❡♥s✐♦♥❛•✱ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✇✐t❤ ❇r♦✇♥✐❛♥ ❜r✐❞❣❡ ✭✶✳✷✮ t♦ t❤❡ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✷✺ ❡q✉❛t✐♦♥ ✭✶✳✺✮✱ ✉s✐♥❣ t❤❡ r❡•❛t✐♦♥ (B (t) = W (t) -t W (1))✱ •❡❛❞s t♦ t❤❡ ❝♦rr❡s✲ ♣♦♥❞✐♥❣ ❋r❡❞❤♦•♠ ❡q✉❛t✐♦♥ ✭s❡❡ ✭✶✳✷✽✮ ❜❡•♦✇✮ ✇✐t❤ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛• g (•, •) ❛♥❞ h (•, •)✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ s♦•✉t✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ ❢♦r♠ ♦❢ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ❜❡❝♦♠❡ ♠✉❝❤ ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞✳ ❚❤✐s ✐s ♣r♦❜❛❜•② t❤❡ r❡❛s♦♥ ✇❤② t❤✐s ♣r♦❜•❡♠ ✇❛s ♥♦t ❝♦♥s✐❞❡r❡❞ t✐•• ♥♦✇✳ ❖✉r r❡s✉•ts t❤✉s s❤♦✉•❞ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛ ❝♦♥str✉❝t✐✈❡ ❡①✐st❡♥❝❡ r❡s✉•t ❢♦r ❆❉❋ t❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼❉❊✳ ✶✳✷ ▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❡❞ ♣r♦❝❡ss X ε = (X t , 0 ≤ t ≤ T ) ✐s t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❞X t = S(X t ) ❞t + ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ✭✶✳✼✮ ✇❤❡r❡ W t , 0 ≤ t ≤ T ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✱ t❤❡ ✐♥✐t✐❛• ✈❛•✉❡ x 0 ✐s ❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ t❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t S(x) ✐s s♦♠❡ ✉♥❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣♦s✐t❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ✇❤❡r❡ ϑ ✐s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ✭✉♥❦♥♦✇♥✮ ♣❛r❛♠❡t❡r✱ ❛❣❛✐♥st ❛•t❡r♥❛t✐✈❡ H 1 : ♥♦t H 0 ✱ ✐✳❡✳✱ t❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t S (x) ✐♥ t❤❡ ♦❜s❡r✈❡❞ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✭✶✳✼✮ ❞♦❡s ♥♦t ❜❡•♦♥❣ t♦ t❤❡ ♣❛r❛♠❡tr✐❝ ❢❛♠✐•② {S (ϑ, x) , ϑ ∈ Θ}✳ ❚❤❡r❡❢♦r❡ t❤❡ ♣r♦❝❡ss ❤❛s t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛•✱ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✱ ❞X t = S(ϑ, X t ) ❞t + ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T. ✭✶✳✽✮ ❍❡r❡ S (ϑ, x) ✐s ❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ t❡st ✐♥ t❤❡ ❛s②♠♣t♦t✐❝s ♦❢ s♠❛•• ♥♦✐s❡✱ ✐✳❡✳✱ ❛s ε → 0✳ ❇❡•♦✇ ❛♥❞ ✐♥ t❤❡ s❡q✉❡• t❤❡ ❞♦t ♠❡❛♥s ❞❡r✐✈❛t✐♦♥ ✇✳r✳t✳ ϑ✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s✳ R. ❚❤❡ ❢✉♥❝t✐♦♥ S(ϑ, x) ✐s str✐❝t•② ♣♦s✐t✐✈❡ ❛♥❞ ❤❛s t✇♦ ❝♦♥t✐♥✉♦✉s ❜♦✉♥❞❡❞ ❞❡✲ r✐✈❛t✐✈❡s ✇✐t❤ r❡s♣❡❝t t♦ ϑ ❛♥❞ x✳ ■♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❜❡•♦✇ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s H 0 ❛r❡ ❛•✇❛②s ❢✉•✜••❡❞✳ ✷✻ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ■t ✐s ❦♥♦✇♥ t❤❛t t❤❡ s♦•✉t✐♦♥ X t ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠•② ✐♥ t ∈ [0, T ] t♦ t❤❡ s♦•✉t✐♦♥ x t = x t (ϑ) ♦❢ t❤❡ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❞x t ❞t = S(ϑ, x t ), x 0 , 0 ≤ t ≤ T, ✇❤❡r❡ x 0 ✐s t❤❡ s❛♠❡ ❛s ✐♥ ✭✶✳✼✮ ✭s❡❡ ❬✷✷( ♦r ❬✸✼(✱ ▲❡♠♠❛ ✶✳✶✸ ❢♦r t❤❡ ♣r♦♦❢✮✳ |❡❝❛•• t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❛①✐♠✉♠ •✐❦❡•✐❤♦♦❞ ❛♥❞ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡s✲ t✐♠❛t♦rs ♦❢ t❤❡ ♣❛r❛♠❡t❡r ϑ✳ ❚❤❡ •✐❦❡•✐❤♦♦❞ r❛t✐♦ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦❜s❡r✲ ✈❛t✐♦♥s ✭✶✳✽✮ ✐s ℓ (ϑ, X ε ) = ❞P ϑ ❞P 0 = exp T 0 S (ϑ, X t ) ε 2 ❞X t - T 0 S (ϑ, X t ) 2 2 ε 2 ❞t , ϑ ∈ Θ, ✇❤❡r❡ P ϑ ❛♥❞ P 0 ❛r❡ t❤❡ ♠❡❛s✉r❡s ✐♥❞✉❝❡❞ r❡s♣❡❝t✐✈❡•② ❜② t❤❡ ♣r♦❝❡ss❡s ✭✶✳✽✮ ❛♥❞ ❞X t = ε ❞W t , X 0 = x 0 , 0 ≤ t ≤ T, ✭s❡❡ ❬✹✽( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮✳ ❚❤❡ ▼▲❊ θε ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ sup ϑ∈Θ ℓ (ϑ, X ε ) = ℓ( θε , X ε ). ❚❤✐s ❡st✐♠❛t♦r ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• ✭❛s ε → 0✮ ε -1 ( θε -ϑ) =⇒ N 0, I (ϑ) -1 , I (ϑ) = T 0 Ṡ (ϑ, x t ) 2 dt. ❍❡r❡ ❛♥❞ ✐♥ t❤❡ s❡q✉❡• x t = x t (ϑ)✳ ❋♦r t❤❡ ♣r♦♦❢ s❡❡ ❑✉t♦②❛♥ts ❬✸✼(✳ ■♥tr♦❞✉❝❡ t❤❡ ▼❉❊ ϑ * ε = arg min ϑ∈Θ X -x(ϑ) , ✇❤❡r❡ • ✐s L 2 [0, T ] ♥♦r♠ ❞❡✜♥❡❞ ❜② X -x(ϑ) 2 = T 0 (X t -x t (ϑ)) 2 ❞t. ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ▼❉❊ ❢♦r t❤✐s ♠♦❞❡• ✇❡r❡ st✉❞✐❡❞ ✐♥ ❬✸✼(✳ ❚❤❡ ▼❉❊ s❛t✐s✜❡s t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡q✉❛t✐♦♥ ✭▼❉❊q✮ T 0 (X t -x t (ϑ * ε )) ẋt (ϑ * ε ) ❞t = 0. ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✷✼ ▲❡t ✉s ♣✉t u * ε = ε -1 (ϑ * ε -ϑ)✳ ❚❤❡r❡❢♦r❡ t❤❡ ▼❉❊q ✐s ❛s ❢♦••♦✇s ✿ T 0 (X t -x t (ϑ + εu * ε )) ẋt (ϑ * ε ) ❞t = 0. ❚❤❡♥✱ ❜② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ ✇❡ ❝❛♥ ✇r✐t❡ T 0 X t -x t (ϑ) -ε u * ε ẋt ( θ) ẋt (ϑ * ε ) ❞t = 0, ✇❤❡r❡ | θ -ϑ| ≤ |ϑ * ε -ϑ| ❛♥❞ ❢♦r u * ε ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ u * ε = J(ϑ) -1 T 0 ε -1 (X t -x t (ϑ)) ẋt (ϑ) ❞t + o(1), ✭✶✳✾✮ ✇❤❡r❡ J(ϑ) = T 0 ẋt (ϑ) 2 ❞t. ✭✶✳✶✵✮ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ♣r♦❝❡ss x (1) t ❞❡✜♥❡❞ ❛s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ X t ✇✳r✳t✳ ε ❛t ε = 0✱ ✐✳❡✳✱ ✇❡ ❤❛✈❡ ε -1 (X t -x t (ϑ)) → x (1) t ✳ ❍❡r❡ t❤❡ r❛♥❞♦♠ ♣r♦❝❡ss x (1) t s❛t✐s✜❡s t❤❡ •✐♥❡❛r ❡q✉❛t✐♦♥ ❞x (1) t = S ′ (ϑ, x t ) x (1)
t ❞t + ❞W t , x (1)
0 = 0, 0 ≤ t ≤ T. ❍❡r❡ ❛♥❞ ❜❡•♦✇ t❤❡ ♣r✐♠❡ ❞❡s✐❣♥❛t❡s ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✳r✳t✳ t❤❡ ✈❛r✐❛❜•❡ x✳ ❚❤❡ ♣r♦❝❡ss x (1) t ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ x (1) t = S(ϑ, x t ) t 0 1 S(ϑ, x s ) ❞W s ✭✶✳✶✶✮ ✭s❡❡✱ ❡✳❣✳✱ ❙❡❝t✐♦♥ 3.3 ✐♥ ❬✸✼(✮✳ ❉✉❡ t♦ t❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ x (1) t , t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✾✮ ❜❡❝♦♠❡s u * ε = J(ϑ) -1 T 0 x (1) t ẋt (ϑ) ❞t + o(1) = J(ϑ) -1 T 0 S(ϑ, x t ) t 0 1 S(ϑ, x s ) ❞W s ẋt (ϑ) ❞t + o(1). ❍❡r❡ o(1) ♠❡❛♥s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t②✱ ✐✳❡✳✱ ❢♦r ❛♥② ν > 0✱ ✇❡ ❤❛✈❡ lim ε→0 P ϑ (|u * ε -ρ(ϑ)| > ν) = 0, ✭✶✳✶✷✮ ✇❤❡r❡ ρ(ϑ) = J(ϑ) -1 T 0 S(ϑ, x t ) t 0 1 S(ϑ, x s ) ❞W s ẋt (ϑ) ❞t. ✷✽ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❚❤❡r❡❢♦r❡✱ ❜② ❋✉❜✐♥✐✬s t❤❡♦r❡♠✱ ϑ * ε ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥✱ ε -1 (ϑ * ε -ϑ) = J(ϑ) -1 T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v + o(1). ✭✶✳✶✸✮ ▼♦r❡♦✈❡r✱ ✉♥❞❡r ❝♦♥❞✐t✐♦♥s ♦❢ r❡❣✉•❛r✐t②✱ t❤❡ ❡st✐♠❛t♦r ϑ * ε ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• ✭s❡❡ ❈❤❛♣t❡r ✼ ✐♥ ❬✸✼(✮ L ϑ {ε -1 (ϑ * ε -ϑ)} =⇒ L{ξ} = N (0, σ 2 (ϑ)), ✇❤❡r❡ σ 2 (ϑ) = J(ϑ) -2 T 0 1 S(ϑ, x v ) 2 T v S(ϑ, x s ) ẋs (ϑ) ❞s 2 ❞v. ✶✳✸ ❇❛s✐❝ st❛t✐st✐❝ ❖✉r ❣♦❛• ✐s t♦ ✜♥❞ t❤❡ •♦❋ t❡st✱ ✇❤✐❝❤ ✐s ❆❉❋✱ ✐✳❡✳✱ ✇❡ s❡❡❦ t❤❡ t❡st st❛t✐st✐❝ ✇❤♦s❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥✱ ✉♥❞❡r ❤②♣♦t❤❡s✐s✱ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❞❡r•②✐♥❣ ♠♦❞❡• ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ S(ϑ, x) ❛♥❞ ♣❛r❛♠❡t❡r ϑ✳ ■♥tr♦❞✉❝❡ t❤❡ st❛t✐st✐❝ δ * ε = T 0 X t -x t (ϑ * ε ) ε S (ϑ * ε , X t ) 2 ❞t. ❚♦ st✉❞② ✐t ✇❡ ♥❡❡❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ X t -x t (ϑ * ε )✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❢♦••♦✇s ✿ ε -1 (X t -x t (ϑ * ε )) = ε -1 (X t -x t (ϑ)) -ε -1 (ϑ * ε -ϑ) ẋt (ϑ) + o(1) = x (1) t (ϑ) -J(ϑ) -1 T 0 x (1) s ẋs (ϑ) ❞s ẋt (ϑ) + o(1) = S(ϑ, x t ) t 0 ❞W v S(ϑ, x v ) -J(ϑ) -1 T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v ẋt (ϑ) + o(1), ✇❤❡r❡ t❤❡ ♣r♦❝❡ss x (1) t ✐s ❞❡✜♥❡❞ ❜② ✭✶✳✶✶✮ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ẋt (ϑ) ✇✳r✳t✳ ϑ s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥ ❞ ẋt (ϑ) ❞t = S ′ (ϑ, x t ) ẋt (ϑ) + Ṡ(ϑ, x t ), ẋ0 (ϑ) = 0. ■ts s♦•✉t✐♦♥ ✐s t❤❡ ❢✉♥❝t✐♦♥ ✭✐t ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✷(✮ ẋt (ϑ) = S(ϑ, x t ) t 0 Ṡ(ϑ, x v ) S(ϑ, x v ) ❞v. ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✷✾ ❍❡r❡ o(1) ✐s t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② ✇✳r✳t✳ t ∈ [0, T ]✱ ✐✳❡✳✱ ❢♦r ❛♥② ν > 0✱ ✇❡ ❤❛✈❡ lim ε→0 P ϑ sup t∈[0,T ] ε -1 (X t -x t (ϑ * ε )) -µ t (ϑ) > ν = 0, ✭✶✳✶✹✮ ✇❤❡r❡ µ t (ϑ) = S(ϑ, x t ) t 0 ❞W v S(ϑ, x v ) -J(ϑ) -1 T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v ẋt (ϑ). ❋♦r t❤❡ ❞❡t❛✐•s s❡❡ ❬✸✼(✱ ❈❤❛♣t❡r ✼✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ t ∈ [0, T ] ✭✐♥ ♣r♦❜❛❜✐•✐t②✮ u ε (t) = X t -x t (ϑ * ε ) ε S (ϑ * ε , X t ) -→ u(t) = t 0 ❞W v S(ϑ, x v ) -J(ϑ) -1 T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v t 0 Ṡ(ϑ, x v ) S(ϑ, x v ) ❞v ✭✶✳✶✺✮ ❛♥❞ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t δ * ε =⇒ T 0 u(t) 2 ❞t, ✭s❡❡ ✐♥ ❬✹✷( t❤❡ ❞❡t❛✐•s ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✇❤❡r❡ t❤❡ ▼▲❊ ✇❛s ✉s❡❞✮✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✐s ♥♦t ❆❉❋✳ ❍❡♥❝❡ t♦ ♦❜t❛✐♥ ❛♥ ❆❉❋ •♦❋ t❡st ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ •❛✉ss✐❛♥ ♣r♦❝❡ss U t T = 1 √ T t 0 S(ϑ, x s ) ❞u(s), 0 ≤ t ≤ T, ✭✶✳✶✻✮ ✇❤❡r❡ u (•) ✐s ❣✐✈❡♥ ❜② ✭✶✳✶✺✮✳ ❚❤❡♥✱ ❜② t❤❡ ■tô ❢♦r♠✉•❛✱ ❞ (S(ϑ, x s )u(s)) = ❞S(ϑ, x s ) u(s) + ❞u(s) S(ϑ, x s ) ❛♥❞ ✉s✐♥❣ t❤❡ ❡q✉❛•✐t② ❞S(ϑ, x s ) = S ′ (ϑ, x s ) S(ϑ, x s ) ❞s, ✭✶✳✶✼✮ ✇❡ ❤❛✈❡ S(ϑ, x t ) u(t) = t 0 S ′ (ϑ, x s ) S(ϑ, x s ) u(s) ❞s + t 0 S(ϑ, x s ) ❞u(s). ✸✵ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❚❤❡r❡❢♦r❡ t❤❡ ♣r♦❝❡ss U (•) ❣✐✈❡♥ ❜② ✭✶✳✶✻✮ ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ (0 ≤ t ≤ T ) U t T = 1 √ T S(ϑ, x t ) u(t) - 1 √ T t 0 S ′ (ϑ, x s ) S(ϑ, x s ) u(s) ❞s. ✭✶✳✶✽✮ ❋✉rt❤❡r ✇❡ ❞❡✜♥❡ t✇♦ ❢✉♥❝t✐♦♥s h(ϑ, r) = T J(ϑ) -1 Ṡ(ϑ, x rT (ϑ)) C (ϑ) 1/2 ✭✶✳✶✾✮ ❛♥❞ g(ϑ, r) = S(ϑ, x rT (ϑ)) -1 1 r S(ϑ, x zT (ϑ)) ẋzT (ϑ) ❞z C (ϑ) -1/2 . ✭✶✳✷✵✮ ❍❡r❡ J(ϑ) = 1 0 ẋvT (ϑ) 2 ❞v ❛♥❞ C (ϑ) = 1 0 S(ϑ, x vT (ϑ)) -2 1 v S(ϑ, x zT (ϑ)) ẋzT (ϑ) ❞z 2 ❞v.
❖❜s❡r✈❡ t❤❛t

1 0 g(ϑ, r) 2 ❞r = 1. ▲❡♠♠❛ ✶✳✸✳✶✳ ❲❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t② U (ν) = W (ν) - 1 0 g(ϑ, r) ❞W (r) ν 0 h(ϑ, r) ❞r, 0 ≤ ν ≤ 1, ✭✶✳✷✶✮ ✇❤❡r❡ W (ν), 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ Pr♦♦❢✳ ❇② t❤❡ ■tô ❢♦r♠✉•❛✱ ✇❡ ❤❛✈❡✱ ❢♦r 0 ≤ t ≤ T ✱ ❞U t T = 1 √ T ❞S(ϑ, x t ) u(t) + 1 √ T S(ϑ, x t ) ❞u(t) - 1 √ T S ′ (ϑ, x t ) S(ϑ, x t ) u(t) ❞t. ❚❤❡♥ ❜② ✭✶✳✶✼✮ ❛♥❞ ✉s✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✶✳✶✺✮ ♦❢ t❤❡ ♣r♦❝❡ss u(•)✱ ✇❡ ♦❜t❛✐♥ U t T = 1 √ T t 0 S(ϑ, x s ) ❞u(s) = W t √ T - J(ϑ) -1 √ T T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v t 0 Ṡ(ϑ, x v ) ❞v. ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✸✶ ❍❡♥❝❡✱ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ν = t T ❛♥❞ W (ν) = T -1/2 W νT , 0 ≤ ν ≤ 1✱ ✇❡ ❤❛✈❡ U (ν) = W (ν) - J(ϑ) -1 √ T T 0 1 S(ϑ, x v ) T v S(ϑ, x s ) ẋs (ϑ) ❞s ❞W v νT 0 Ṡ(ϑ, x v ) ❞v. ▲❡t ✉s ❝❤❛♥❣❡ t❤❡ ✈❛r✐❛❜•❡s r = v T ✱ z = s T ✱ W (r) = T -1/2 W rT , 0 ≤ r ≤ 1✳ ❚❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ U (ν) = W (ν) -J(ϑ) -1 1 0 1 S(ϑ, x rT ) T rT S(ϑ, x s ) ẋs (ϑ) ❞s ❞W (r) ν 0 Ṡ(ϑ, x rT ) ❞r = W (ν) - T J(ϑ) 1 0 1 S(ϑ, x rT ) 1 r S(ϑ, x zT ) ẋzT (ϑ) ❞z ❞W (r) ν 0 Ṡ(ϑ, x rT ) ❞r = W (ν) - 1 0 g(ϑ, r) ❞W (r) ν 0 h(ϑ, r) ❞r. ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✶✳✷✶✮ ❛♥❞ t❤✐s ♣r♦✈❡s t❤❡ ▲❡♠♠❛ ✶✳✸✳✶✳ ❯s✐♥❣ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ U ε (•) t♦ U (•) ✭s❡❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳✺✳✷ ❜❡•♦✇✮ ❛♥❞ ❞✉❡ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ t❤❡♦r❡♠✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∆ ε = 1 T 2 T 0 t 0 S(ϑ * ε , X s ) ❞u ε (s) 2 ❞t =⇒ 1 0 U (ν) 2 ❞ν. ❲❡ r❡♠❛r❦ t❤❛t t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✐s ♥♦t ❆❉❋✳ ❍❡♥❝❡ ✇❡ ❤❛✈❡ t♦ ✜♥❞ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L[U ](•) ✐♥t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss s✉❝❤ t❤❛t 1 0 L[U ](ν) 2 ❞ν = 1 0 w 2 ν ❞ν, 0 ≤ ν ≤ 1, ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤✐s ♣r♦♣❡rt② ❛••♦✇s ✉s t♦ ♣r❡s❡♥t t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✇✐t❤ t❤❡ s❛♠❡ •✐♠✐t✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✐s ❆❉❋✳ ✶✳✹ ▲✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ◆♦✇ t❤❡ ♣r♦❜•❡♠ ✐s t♦ ✜♥❞ s✉❝❤ tr❛♥s❢♦r♠❛t✐♦♥ L [•] ♦❢ U (•) ✭s❡❡ ✭✶✳✷✶✮✮ ✉s✐♥❣ t❤❡ ▼❉❊ t❤❛t L[U ](ν) = w ν . |❡❝❛•• t❤❛t ❢♦r t❤❡ •✐♠✐t ♣r♦❝❡ss ✭✶✳✶✮ s✉❝❤ •✐♥❡❛r ✸✷ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❆❉❋ t❡st ✇❡r❡ ♣r♦♣♦s❡❞ ❜② ❑❤♠❛•❛❞③❡ ❬✸✵(✳ ❆♥♦t❤❡r ✭❞✐r❡❝t✮ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉•t ✇❛s r❡❝❡♥t•② ♦❜t❛✐♥❡❞ ❜② ❑•❡♣ts②♥❛ ❛♥❞ ❑✉t♦②❛♥ts ❬✸✹(✳ ◆♦t❡ t❤❛t ✐♥ t❤❡s❡ ✇♦r❦s t❤❡ ❡st✐♠❛t♦r ✉s❡❞ ✇❛s ❛•✇❛②s t❤❡ ▼▲❊ ❛♥❞ ✐♥ ♦✉r ✇♦r❦ ✐t ✐s t❤❡ ▼❉❊✳ ❚❤❡ •✐♠✐t ♣r♦❝❡ss❡s ✐♥ t❤❡s❡ t✇♦ ❝❛s❡s ❛r❡ q✉✐t❡ ❞✐✛❡r❡♥t✳ ❚❤❛t ✐s ✇❤② ✇❡ ❤❛✈❡ t♦ ♣r❡s❡♥t ❤❡r❡ ❛ s♣❡❝✐❛• ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦♦❢ ❣✐✈❡♥ ✐♥ ❬✸✹(✳ ❖✉r ♣r♦♦❢ ❢♦••♦✇s t❤❡ ♠❛✐♥ st❡♣s ♦❢ t❤❡ ✇♦r❦ ❬✸✹(✳ ❙♣❡❝✐✜❝❛••②✱ ✇❡ ❤❛✈❡ t♦ s♦•✈❡ ❋r❡❞❤♦•♠ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ✇✐t❤ ❞❡❣❡♥❡r❛t❡❞ ❦❡r♥❡•✳ ❚❤❡ s♦•✉t✐♦♥ ♦❢ ✐t ❣✐✈❡s ✉s t❤❡ ❞❡s✐r❡❞ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ❉❡♥♦t❡ I 1 = r 0 g(ϑ, q) 2 ❞q, I 2 = r 0 h(ϑ, q) g(ϑ, q) ❞q, I 3 = r 0 h(ϑ, q) ❞q, I 4 = r 0 h(ϑ, q) 2 ❞q, I 5 = r 0 g(ϑ, q) ❞q. ✭✶✳✷✷✮ ❇❡•♦✇ ✇❡ ♦♠✐t ϑ ❛♥❞ r ❢♦r s✐♠♣•✐❝✐t② ❛♥❞ ♣✉t g = g(ϑ, r) ❛♥❞ h = h(ϑ, r)✳ ■♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥s ϕ 1 (r) = g -h -3I 2 g + I 5 hg + I 3 g 2 + 2I 2 h -2I 2 I 3 g 2 + I 1 I 2 2 h + I 4 I 5 g 2 -I 3 2 g -I 2 I 4 g + 3I 2 2 g + I 2 I 5 h 2 -2I 2 I 5 hg -2I 1 I 2 h + I 2 2 I 5 hg + I 2 1 I 3 h 2 -I 4 h + 2I 1 I 4 h -I 1 I 4 g + I 1 I 2 I 4 g + I 1 I 4 I 5 hg -I 2 1 I 4 h + I 1 h + 2I 2 I 3 hg -I 2 I 4 I 5 g 2 -I 5 h 2 + 2I 1 I 3 hg -2I 1 I 2 I 3 hg -2I 1 I 3 h 2 -I 2 2 h + I 3 h 2 -2I 3 hg -I 4 I 5 hg -I 1 I 2 I 5 h 2 + I 2 2 I 3 g 2 + I 1 I 5 h 2 + I 4 g, ✭✶✳✷✸✮ ϕ 2 (r) = 1 + I 5 h -3I 2 I 5 h + I 1 I 3 h + I 3 g -3I 2 I 3 g + I 4 I 5 g -I 3 h -I 1 I 2 4 I 5 g + 3I 2 2 I 3 g -2I 2 I 4 I 5 g + 2I 2 I 3 h -I 3 2 I 5 h + I 1 I 2 2 I 3 h -I 3 2 I 3 g + 3I 2 2 I 5 h + I 4 I 5 h -I 2 I 4 I 5 h + 2I 1 I 3 I 4 h + I 3 I 4 g -I 2 I 3 I 4 g + I 2 4 + 2I 2 2 I 4 + I 4 2 + I 1 I 2 I 4 I 5 h -I 2 1 I 3 I 4 h -I 1 I 3 I 4 g + I 1 I 2 I 3 I 4 g -I 2 2 I 3 h -2I 1 I 2 2 I 4 -2I 1 I 2 4 + I 2 1 I 2 4 + 2I 4 -2I 1 I 4 -4I 2 I 4 + 4I 1 I 2 I 4 -4I 2 + I 2 4 I 5 g + I 2 2 I 4 I 5 g -2I 1 I 2 I 3 h -I 3 I 4 h + 6I 2 2 -4I 3 2 -I 1 I 4 I 5 h ✭✶✳✷✹✮ ❛♥❞ ψ 2 (r) = h + I 3 hg -2I 2 I 4 g + I 5 h 2 -3I 2 h -2I 2 I 3 hg -I 1 I 2 I 3 h 2 + I 4 g + 3I 2 2 h ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✸✸ + I 2 2 I 4 g + I 2 2 I 5 h 2 -I 3 2 h -I 3 I 4 hg + I 2 4 g + I 4 h -I 2 I 4 h -2I 2 I 5 h 2 -I 1 I 4 h + I 1 I 2 I 4 h + I 1 I 3 h 2 + I 3 I 4 g 2 + I 2 I 3 h 2 -I 3 h 2 + I 1 I 3 I 4 hg -I 2 I 3 I 4 g 2 -2I 2 I 4 I 5 hg + 2I 4 I 5 hg + I 2 2 I 3 hg -I 1 I 2 4 g + I 2 4 I 5 g 2 . ✭✶✳✷✺✮ ❚❤❡ ❢♦••♦✇✐♥❣ ❚❤❡♦r❡♠ ✐s t❤❡ ♠❛✐♥ r❡s✉•t ♦❢ t❤✐s ✇♦r❦✳ ❚❤❡♦r❡♠ ✶✳✹✳✶✳ ❙✉♣♣♦s❡ t❤❛t h(q) ❛♥❞ g(q) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t 1 0 g(q) 2 ❞q = 1 ❛♥❞ ϕ 2 (r) ✐s ❛ str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ [0, 1)✳ ❚❤❡♥ t❤❡ ❡q✉❛✲ •✐t② L[U ](ν) = U (ν) + ν 0 r 0 ϕ 1 (r) h(q) + ψ 2 (r) g(q) ϕ 2 (r) ❞U (q) ❞r = w ν ✭✶✳✷✻✮ ❤♦•❞s✳ ❍❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ✇✐•• ❜❡ ❞♦♥❡ ✐♥ s❡✈❡r❛• st❡♣s✳ ❙t❡♣ ✶ ✿ ■♥tr♦❞✉❝❡ ❛ •❛✉ss✐❛♥ ♣r♦❝❡ss M t = t 0 q(t, s) ❞U (s), 0 ≤ t ≤ 1, ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ q(t, s) ✐s ❝❤♦s❡♥ ❛s s♦•✉t✐♦♥ ♦❢ ❋r❡❞❤♦•♠ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♥❡①t st❡♣✳ ❖❜s❡r✈❡ t❤❛t M t = t 0 q(t, u) ❞W (u) - 1 0 g(u) ❞W (u) t 0 q(t, u) h(u) ❞u. ❙t❡♣ ✷ ✿ ❋♦r t❤❡ ❝♦rr❡•❛t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ M t R(t, s) = ❊ [M t M s ] , t > s, ✇❡ ❤❛✈❡ ❊ [M t M s ] = ❊ t 0 q(t, u) ❞W (u) - 1 0 g(u) ❞W (u) t 0 q(t, u) h(u) ❞u s 0 q(s, v) ❞W (v) - 1 0 g(v) ❞W (v) s 0 q(s, v) h(v) ❞v = s 0 q(t, u) q(s, u) ❞u - t 0 q(t, u) g(u) ❞u s 0 q(s, v) h(v) ❞v - s 0 q(s, v) g(v) ❞v t 0 q(t, u) h(u) ❞u + 1 0 g(v) 2 ❞v =1 t 0 q(t, u) h(u) ❞u s 0 q(s, v) h(v) ❞v ✸✹ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ = s 0 q(s, u) q(t, u) - t 0 q(t, v) g(v) ❞v h(u) - t 0 q(t, v) h(v) ❞v g(u) + t 0 q(t, v) h(v) ❞v h(u) ❞u = s 0 q(s, u) q(t, u) - t 0 q(t, v) g(v) h(u) + h(v) g(u) -h(v) h(u) ❞v ❞u. ❉❡♥♦t❡ t❤❡ ❦❡r♥❡• K(u, v) = g(v) h(u) + h(v) g(u) -h(v) h(u). ❚❤❡♥ ❊[M t M s ] = s 0 q(s, u) q(t, u) - t 0 q(t, v) K(u, v) ❞v ❞u. ✭✶✳✷✼✮ ❚❤❡r❡❢♦r❡ ✐❢ ✇❡ t❛❦❡ q(t, s) s✉❝❤ t❤❛t ✐t s♦•✈❡s t❤❡ ❋r❡❞❤♦•♠ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ (t ✐s ✜①❡❞) q(t, s) - t 0 q(t, v) K(s, v) ❞v = 1, s ∈ [0, t], ✭✶✳✷✽✮ t❤❡♥ (1.27) ❜❡❝♦♠❡s ❊[M t M s ] = ❊ M 2 s = s 0 q(s, u) ❞u. ✭✶✳✷✾✮ ❙t❡♣ ✸ ✿ ❚❤❡ s♦•✉t✐♦♥ q(t, s) ❝❛♥ ❜❡ ❢♦✉♥❞ ❛s ❢♦••♦✇s ✿ ❲❡ ❤❛✈❡ q(t, s) = 1 + t 0 q(t, v) K(s, v) ❞v. ❉❡♥♦t❡ A(t) = t 0 q(t, v) h(v) ❞v ❛♥❞ B(t) = t 0 q(t, v) g(v) ❞v. ❚❤❡♥ q(t, s) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ q(t, s) = 1 + B(t) h(s) + A(t) (g(s) -h(s)) , ✭✶✳✸✵✮ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✸✺ ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ A(t) ✐ts❡•❢ ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛t✐♦♥ ✭❛❢t❡r ♠✉•t✐✲ ♣•②✐♥❣ ✭✶✳✸✵✮ ❜② h(s) ❛♥❞ ✐♥t❡❣r❛t✐♥❣✮ t 0 h(s) ❞s = A(t) -B(t) t 0 h(s) 2 ❞s -A(t) t 0 h(s) g(s) ❞s + A(t) t 0 h(s) 2 ❞s. ✭✶✳✸✶✮ ❚❤❡ ❢✉♥❝t✐♦♥ B(t) ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛t✐♦♥ ✭❛❢t❡r ♠✉•t✐♣•②✐♥❣ ✭✶✳✸✵✮ ❜② g(s) ❛♥❞ ✐♥t❡❣r❛t✐♥❣✮ t 0 g(s) ❞s = B(t) -B(t) t 0 h(s) g(s) ❞s -A(t) t 0 g(s) 2 ❞s + A(t) t 0 h(s) g(s) ❞s. ✭✶✳✸✷✮ ❯s✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ✭✶✳✷✷✮✱ ✇❡ ❝❛♥ ✇r✐t❡ ✭✶✳✸✶✮-✭✶✳✸✷✮ ❛s ❢♦••♦✇s ✿ A(t) -B(t) I 4 -A(t) I 2 + A(t) I 4 = I 3 ✭✶✳✸✸✮ ❛♥❞ B(t) -B(t) I 2 -A(t) I 1 + A(t) I 2 = I 5 . ✭✶✳✸✹✮ ❋✉rt❤❡r✱ ✇❡ ❤❛✈❡ t♦ ✜♥❞ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ A(t) ❛♥❞ B(t). ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ ❢r♦♠ ✭✶✳✸✹✮ B(t) = I 5 + A(t) (I 1 -I 2 ) (1 -I 2 ) . ✭✶✳✸✺✮ ❚❤❡♥ ✇❡ ✐♥s❡rt ✭✶✳✸✺✮ ✐♥ ✭✶✳✸✸✮ ❛♥❞ ♦❜t❛✐♥ A(t) = I 3 (1 -I 2 ) + I 4 I 5 (1 -I 2 ) 2 + I 4 I 6 , B(t) = I 5 (1 + I 4 -I 2 ) + I 3 (I 1 -I 2 ) (1 -I 2 ) 2 + I 4 I 6 , ✇❤❡r❡ I 6 = 1 t g(s) 2 ❞s. ❚❤❡r❡❢♦r❡ t❤❡ s♦•✉t✐♦♥ q(t, s) ♦❢ ✭✶✳✷✽✮ ✐s q(t, s) = 1 + I 5 (1 + I 4 -I 2 ) + I 3 (I 1 -I 2 ) (1 -I 2 ) 2 + I 4 I 6 h(s) + I 3 (1 -I 2 ) + I 4 I 5 (1 -I 2 ) 2 + I 4 I 6 (g(s) -h(s)). ❚❤❡ ✜♥❛• ❡①♣r❡ss✐♦♥ ♦❢ q(t, s) ✐s q(t, s) = 1 + t 0 g(s)❞s 1 + t 0 h(s) 2 ❞s - t 0 h(s)g(s) ❞s h(s) 1 - t 0 h(s)g(s) ❞s 2 + t 0 h(s) 2 ❞s 1 t g(s) 2 ❞s ✸✻ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ + t 0 h(s)❞s t 0 g(s) 2 ❞s - t 0 h(s)g(s) ❞s h(s) 1 - t 0 h(s)g(s) ❞s 2 + t 0 h(s) 2 ❞s 1 t g(s) 2 ❞s + t 0 h(s)❞s 1 - t 0 h(s)g(s)❞s + t 0 h(s) 2 ❞s t 0 g(s)❞s 1 - t 0 h(s)g(s)❞s 2 + t 0 h(s) 2 ❞s 1 t g(s) 2 ❞s (g(s) -h(s)). ❙t❡♣ ✹ ✿ ❚♦ s❤♦✇ t❤❛t M t ✐s ♠❛rt✐♥❣❛•❡ ✇❡ ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ ▲❡♠♠❛✳ ▲❡♠♠❛ ✶✳✹✳✷✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛•✐t② t 0 q(t, s) ❞s = t 0 q(s, s) 2 ❞s, 0 ≤ t ≤ 1. ✭✶✳✸✻✮ Pr♦♦❢✳ ❙❤♦✇ t❤❛t ❞ ❞t t 0 q(t, s) ❞s = ❞ ❞t t 0 q(s, s) 2 ❞s = q(t, t) 2 . ❉❡♥♦t❡ C(t) = 1 - t 0 h(s) g(s) ❞s, D(t) = t 0 g(s) ❞s 1 + t 0 h(s) 2 ❞s - t 0 h(s) g(s) ❞s + t 0 h(s) ❞s t 0 g(s) 2 ❞s - t 0 h(s) g(s) ❞s , K(t) = C(t) 2 + t 0 h(s) 2 ❞s 1 t g(s) 2 ❞s ❛♥❞ N (t) = C(t) t 0 h(s) ❞s + t 0 h(s) 2 ❞s t 0 g(s) ❞s. ❚❤❡♥ q(t, s) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ q(t, s) = 1 + D(t) K(t) h(s) + N (t) K(t) (g(s) -h(s)). ❍❡♥❝❡ ❞ ❞t t 0 q(t, s) ❞s = 1 + D(t) h(t) + N (t) (g(t) -h(t)) K(t) + D ′ (t) K(t) -D(t) K ′ (t) K(t) 2 t 0 h(s) ❞s + N ′ (t)K(t) -N (t)K ′ (t) K(t) 2 t 0 (g(s) -h(s)) ❞s. ✭✶✳✸✼✮ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✸✼ ❚❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛•✐t✐❡s C(t) = 1 -I 2 , D(t) = I 5 (1 + I 4 -I 2 ) + I 3 (I 1 -I 2 ) , K(t) = (1 -I 2 ) 2 + I 4 (1 -I 1 ) , N (t) = I 3 (1 -I 2 ) + I 4 I 5 . ❚❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ✇✳r✳t✳ t ❤❛✈❡ s✉❝❤ ❡①♣r❡ss✐♦♥s C ′ (t) = -h(t) g(t), D ′ (t) = g(t) + I 4 g(t) -I 2 g(t) + I 5 h(t) 2 -I 5 h(t) g(t) + I 1 h(t) -I 2 h(t) + I 3 g(t) 2 -I 3 h(t) g(t), K ′ (t) = -2h(t) g(t) + 2I 2 h(t)g(t) + h(t) 2 -I 1 h(t) 2 -I 4 g(t) 2 , N ′ (t) = -I 3 h(t) g(t) + I 4 g(t) + I 5 h(t) 2 + h(t) -I 2 h(t). ❚❤✉s✱ ✭✶✳✸✼✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ❞ ❞t t 0 q(t, s) ❞s = 1 + Φ 1 (t) + Φ 2 (t) + Φ 3 (t) K(t) 2 , ✭✶✳✸✽✮ ✇❤❡r❡ Φ 1 (t) = D(t) h(t) + N (t) (g(t) -h(t)) K(t), Φ 2 (t) = D ′ (t) K(t) -D(t) K ′ (t) I 3 ❛♥❞ Φ 3 (t) = N ′ (t) K(t) -N (t) K ′ (t) (I 5 -I 3 ) . ❚❤❡♥ r❡t✉r♥✐♥❣ t♦ ✐♥✐t✐❛• ♥♦t❛t✐♦♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❡①♣r❡ss✐♦♥s Φ 1 (t) = 3I 2 2 I 5 h -3I 2 I 5 h + I 1 I 3 h -3I 2 I 3 g + I 1 I 2 2 I 3 h -I 3 2 I 5 h + I 3 g -I 3 2 I 3 g + I 4 I 5 g -I 1 I 3 I 4 g -I 3 h -I 2 1 I 3 I 4 h + 2I 2 I 3 h + I 5 h -I 1 I 2 4 I 5 g -I 2 I 3 I 4 g -2I 1 I 2 I 3 h + 3I 2 2 I 3 g -2I 2 I 4 I 5 g + I 4 I 5 h -I 2 I 4 I 5 h + I 1 I 2 I 3 I 4 g + I 2 4 I 5 g + I 2 2 I 4 I 5 g -I 2 2 I 3 h + 2I 1 I 3 I 4 h -I 3 I 4 h -I 1 I 4 I 5 h + I 3 I 4 g + I 1 I 2 I 4 I 5 h, Φ 2 (t) = I 2 I 2 3 h 2 + 2I 3 I 4 g -3I 2 I 3 g + I 3 I 5 hg + I 3 I 2 4 I 5 g 2 + I 2 3 I 4 g 2 + I 1 I 3 h -3I 2 I 3 I 4 g + 3I 2 2 I 3 g -I 2 I 3 I 5 h 2 -2I 2 I 3 I 5 hg + I 2 3 g 2 -I 1 I 2 3 h 2 ✸✽ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ + I 2 2 I 3 I 4 g -I 3 2 I 3 g + I 2 2 I 3 I 5 h 2 + I 2 2 I 3 I 5 hg + I 1 I 2 2 I 3 h -2I 1 I 2 I 3 h -I 2 3 hg -2I 2 I 2 3 g 2 + I 2 2 I 2 3 g 2 + 2I 2 2 I 3 h -2I 2 I 3 I 4 I 5 hg -I 2 1 I 3 I 4 h + I 2 2 I 2 3 hg + I 3 I 2 4 g + I 3 I 4 I 5 hg -2I 1 I 2 I 2 3 hg + I 2 1 I 2 3 h 2 -I 1 I 2 I 2 3 h 2 -I 1 I 3 I 4 g -I 1 I 3 I 2 4 g + I 1 I 2 I 3 I 4 g + I 1 I 3 I 4 I 5 hg -I 1 I 2 I 3 I 5 h 2 + I 3 g + I 1 I 2 3 I 4 hg + I 1 I 3 I 5 h 2 + I 3 I 4 I 5 g 2 + I 1 I 3 I 4 h -I 2 I 3 h -I 2 I 2 3 I 4 g 2 -I 2 I 3 I 4 I 5 g 2 + 2I 1 I 2 3 hg -I 2 3 I 4 hg + I 1 I 2 I 3 I 4 h -I 2 I 3 I 4 h -I 3 2 I 3 h ❛♥❞ Φ 3 (t) = I 3 I 5 hg + I 2 I 3 I 4 h -I 3 I 5 h 2 -3I 2 I 5 h -I 3 I 2 4 g -I 2 2 I 2 3 hg -2I 2 I 2 5 h 2 + 3I 2 2 I 5 h + I 2 2 I 3 I 5 hg -I 2 2 I 3 I 5 h 2 + I 2 2 I 2 5 h 2 -I 3 I 2 4 I 5 g 2 + I 2 4 I 5 g -3I 2 2 I 3 h -I 2 I 4 I 5 h + I 1 I 3 I 4 I 5 hg -I 2 3 hg -I 1 I 2 4 I 5 g + I 2 I 2 3 I 4 g 2 + I 1 I 3 I 5 h 2 + I 3 I 4 I 5 g 2 + I 2 I 3 I 5 h 2 + I 5 h -2I 2 I 3 I 5 hg + 2I 4 I 2 5 hg -2I 2 I 4 I 2 5 hg + I 2 4 I 2 5 g 2 + I 2 2 I 4 I 5 g -I 3 h + I 1 I 2 I 4 I 5 h + 3I 2 I 3 h + 2I 2 I 2 3 hg + 2I 2 I 3 I 4 g + 2I 2 I 3 I 5 h 2 + I 2 5 h 2 -I 1 I 4 I 5 h -I 2 2 I 3 I 4 g + I 3 2 I 3 h + I 2 3 I 4 hg + I 4 I 5 h -I 3 2 I 5 h + I 2 3 h 2 + I 4 I 5 g -I 1 I 2 3 I 4 hg + I 1 I 2 I 2 3 h 2 + I 1 I 3 I 2 4 g + I 1 I 3 I 4 h -I 3 I 4 h -I 3 I 4 g -I 2 3 I 4 g 2 -I 2 I 2 3 h 2 -I 3 I 5 h 2 -I 1 I 2 I 3 I 4 h -I 2 I 3 I 4 I 5 g 2 -I 1 I 2 3 h 2 -2I 3 I 4 I 5 hg + 2I 2 I 3 I 4 I 5 hg -2I 2 I 4 I 5 g -I 1 I 2 I 3 I 5 h 2 -I 3 I 4 I 5 hg. ❉❡♥♦t❡ Φ(t) = Φ 1 (t) + Φ 2 (t) + Φ 3 (t), t❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ Φ(t) = h(t) 2I 5 -6I 2 I 5 + 2I 1 I 3 -2I 3 + 6I 2 2 I 5 -4I 1 I 2 I 3 + 4I 2 I 3 + 4I 1 I 3 I 4 -2I 3 2 I 5 + 2I 1 I 2 2 I 3 + 2I 4 I 5 -2I 2 I 4 I 5 -2I 3 I 4 + 2I 1 I 2 I 4 I 5 -2I 2 1 I 3 I 4 -2I 1 I 4 I 5 -2I 2 2 I 3 + g(t) 2I 3 -2I 1 I 3 I 4 + 2I 1 I 2 I 3 I 4 -6I 2 I 3 + 2I 4 I 5 -2I 1 I 2 4 I 5 + 6I 2 2 I 3 -4I 2 I 4 I 5 -2I 3 2 I 3 + 2I 2 2 I 4 I 5 + 2I 3 I 4 + 2I 2 4 I 5 -2I 2 I 3 I 4 + h(t)g(t) 2I 3 I 5 -2I 2 3 -4I 2 I 3 I 5 -2I 3 I 4 I 5 + 2I 1 I 3 I 4 I 5 + 2I 2 I 2 3 + 2I 4 I 2 5 + 2I 1 I 2 3 -2I 1 I 2 I 2 3 -2I 2 I 4 I 2 5 + 2I 2 2 I 3 I 5 + h(t) 2 ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✸✾ 2I 1 I 3 I 5 -2I 1 I 2 I 3 I 5 -2I 2 I 2 5 + I 2 2 I 2 5 -2I 3 I 5 + 2I 2 I 3 I 5 -2I 1 I 2 3 + I 2 3 + I 2 5 + I 2 1 I 2 3 + g(t) 2 I 2 3 -2I 2 I 2 3 + 2I 3 I 4 I 5 -2I 2 I 3 I 4 I 5 + I 2 4 I 2 5 + I 2 2 I 2 3 . ❋✐♥❛••②✱ ✭✶✳✸✽✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ ❞ ❞t t 0 q(t, s) ❞s = 1 + Φ(t) K(t) 2 . ✭✶✳✸✾✮ ◆♦✇ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ q(t, t) 2 ✐s q(t, t) 2 = 1 + D(t) K(t) h(t) + N (t) K(t) (g(t) -h(t)) 2 = 1 + 2 h(t) K(t) D(t) -N (t) + 2 g(t) K(t) N (t) K(t) 2 + h(t) 2 D(t) -N (t) 2 + g(t) 2 N (t) 2 K(t) 2 + 2 h(t) g(t)N (t) D(t) -N (t) K(t) 2 . ❉❡♥♦t❡ M (t) = 2 K(t) (D(t) -N (t)) , Q(t) = 2 K(t) N (t), L(t) = D(t) -N (t) 2 , E(t) = N (t) 2 , H(t) = 2 N (t) D(t) -N (t) . ❚❤❡r❡❢♦r❡ t❤❡ ✜♥❛• ❡①♣r❡ss✐♦♥ ❢♦r q(t, t) 2 ✐s q(t, t) 2 = 1 + h(t) M (t) + g(t) Q(t) + h(t) 2 L(t) K(t) 2 + g(t) 2 E(t) + h(t) g(t) H(t) K(t) 2 , ✭✶✳✹✵✮ ✇❤❡r❡ M (t) = 2I 1 I 3 -6I 2 I 5 -2I 3 + 6I 2 2 I 5 -4I 1 I 2 I 3 + 4I 2 I 3 -2I 3 2 I 5 + 2I 4 I 5 + 2I 1 I 2 2 I 3 -2I 2 1 I 3 I 4 -2I 2 2 I 3 -2I 2 I 4 I 5 + 2I 5 -2I 3 I 4 -2I 1 I 4 I 5 + 2I 1 I 2 I 4 I 5 + 4I 1 I 3 I 4 , Q(t) = 2I 3 I 4 -6I 2 I 3 + 6I 2 2 I 3 -2I 3 2 I 3 + 2I 4 I 5 -4I 2 I 4 I 5 + 2I 2 2 I 4 I 5 ✹✵ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ + 2I 3 -2I 2 I 3 I 4 -2I 1 I 3 I 4 + 2I 1 I 2 I 3 I 4 + 2I 2 4 I 5 -2I 1 I 2 4 I 5 , L(t) = I 2 5 + I 2 2 I 2 5 -2I 2 I 2 5 + I 2 1 I 2 3 + I 2 3 -2I 1 I 2 3 + 2I 1 I 3 I 5 -2I 3 I 5 + 2I 2 I 3 I 5 -2I 1 I 2 I 3 I 5 , E(t) = I 2 3 -2I 2 I 2 3 + I 2 2 I 2 3 + I 2 4 I 2 5 + 2I 3 I 4 I 5 -2I 2 I 3 I 4 I 5 , H(t) = 2I 3 I 5 -4I 2 I 3 I 5 -2I 3 I 4 I 5 + 2I 1 I 2 3 + 2I 2 I 2 3 + 2I 2 2 I 3 I 5 -2I 2 I 4 I 2 5 + 2I 1 I 3 I 4 I 5 -2I 2 3 -2I 1 I 2 I 2 3 + 2I 4 I 2 5 . ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡s❡ ❡①♣r❡ss✐♦♥s ✇✐t❤ ✭✶✳✸✾✮-✭✶✳✹✵✮ s❤♦✇s t❤❛t t❤❡ ▲❡♠♠❛ ✐s ♣r♦✈❡❞✳ ❙t❡♣ ✺ ✿ ■♥ t❤❡ ♥❡①t st❡♣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦••♦✇✐♥❣ ▲❡♠♠❛ t♦ s❤♦✇ t❤❛t t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ▲❡♠♠❛ ✶✳✹✳✸✳ ■❢ t❤❡ •❛✉ss✐❛♥ ♣r♦❝❡ss M s s❛t✐s✜❡s ✭✶✳✷✾✮ ❛♥❞ ✇❡ ❤❛✈❡ r❡•❛t✐♦♥ ✭✶✳✸✻✮✱ t❤❡♥ z(t) = t 0 q(s, s) -1 dM s ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ Pr♦♦❢✳ ❚❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ❢♦✉♥❞✱ ❡✳❣✳✱ ✐♥ ❬✸✹(✱ ▲❡♠♠❛ ✷✳ ❍❡♥❝❡ M t = t 0 q(s, s) ❞w s , 0 ≤ t ≤ 1 ✐s ❛ •❛✉ss✐❛♥ ♠❛rt✐♥❣❛•❡✱ ✇❤❡r❡ w s , 0 ≤ s ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t② ✇ t = t 0 q(s, s) -1 ❞M s = U (t) + t 0 q(s, s) -1 s 0 q ′ s (s, v) ❞U (v) ❞s, ✇❤❡r❡ w t , 0 ≤ t ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✭❜② ▲❡♠♠❛ ✶✳✹✳✸✮✳ ◆♦✇ ✇❡ ❤❛✈❡ t♦ ❝❛•❝✉•❛t❡ t❤❡ r✐❣❤t s✐❞❡ ♦❢ t❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ q ′ t (t, s) ✇✳r✳t✳ t ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ q ′ t (t, s) = (ψ 1 (t) -ψ 2 (t)) h(s) + ψ 2 (t) g(s) K(t) 2 , ✇❤❡r❡ ψ 1 (t) = D ′ (t) K(t) -D(t) K ′ (t) ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✹✶ ❛♥❞ ψ 2 (t) = N ′ (t) K(t) -N (t) K ′ (t). |❡t✉r♥✐♥❣ t♦ t❤❡ ✐♥✐t✐❛• ♥♦t❛t✐♦♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ✿ ψ 1 (t) = g + 2I 4 g -3I 2 g + I 5 hg + I 1 h -I 2 h + I 1 I 4 I 5 hg + I 2 4 g + 3I 2 2 g -I 2 I 5 h 2 -2I 2 I 5 hg -2I 1 I 2 h + 2I 2 2 h + I 2 2 I 3 g 2 -I 1 I 2 I 3 h 2 -I 3 2 g + I 2 2 I 5 hg + I 1 I 2 2 h -I 3 2 h -2I 2 I 4 I 5 hg + I 2 1 I 3 h 2 -I 1 I 2 I 5 h 2 -I 3 hg -I 2 I 4 h + I 3 I 4 g 2 -I 3 I 4 hg -I 1 I 4 g -I 1 I 2 4 g + I 1 I 2 I 4 g -2I 2 I 3 g 2 + I 1 I 2 I 4 h + I 1 I 5 h 2 + I 4 I 5 g 2 + I 2 4 g + I 2 2 I 3 hg -I 1 I 3 h 2 + I 2 4 I 5 g 2 -I 2 I 4 I 5 g 2 + 2I 1 I 3 hg -2I 1 I 2 I 3 hg + I 2 2 I 4 g -I 2 1 I 4 h + I 2 I 3 h 2 -I 2 I 3 I 4 g 2 -3I 2 I 4 g + I 2 2 I 5 h 2 + I 1 I 4 h + I 1 I 3 I 4 hg + I 4 I 5 hg + I 3 g 2 . ❚❤❡ ❢✉♥❝t✐♦♥ ψ 2 (t) ✐s ❞❡✜♥❡❞ ❜② ✭✶✳✷✺✮✳ ❍❡♥❝❡ ✇❡ ♦❜t❛✐♥ q ′ t (t, s) q(t, t) = (ψ 1 (t) -ψ 2 (t)) h(s) + ψ 2 (t) g(s) K(t) 2 + Φ 1 (t) . ❚❤❡♥ ✐❢ ✇❡ ♣✉t ϕ 1 (t) = ψ 1 (t) -ψ 2 (t) ❛♥❞ ϕ 2 (t) = K(t) 2 + Φ 1 (t), t❤❡♥ t❤✐s ✐♠♣•✐❡s t❤❛t q ′ t (t, s) q(t, t) = ϕ 1 (t) h(s) + ψ 2 (t) g(s) ϕ 2 (t) , ✇✐t❤ ϕ 1 (t) ❛♥❞ ϕ 2 (t) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✶✳✷✸✮-✭✶✳✷✹✮✳ ❋✐♥❛••②✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❡①♣r❡s✲ s✐♦♥ w t = t 0 q(s, s) -1 ❞M s = U (t) + t 0 s 0 ϕ 1 (s) h(v) + ψ 2 (s) g(v) ϕ 2 (s) ❞U(v) ❞s. ❚❤✐s ✐s t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L[U ](•) ♦❢ t❤❡ ♣r♦❝❡ss U (•) ✐♥t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss w t ❛♥❞ t❤✐s ♣r♦✈❡s t❤❡ ❚❤❡♦r❡♠ ✶✳✹✳✶✳ |❡♠❛r❦✳ ▲❡t ✉s ♣r❡s❡♥t ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ϕ 2 (t) > 0✳ ✹✷ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ R 0 . ❙✉♣♣♦s❡ t❤❛t h(t) ❛♥❞ g(t) ❛r❡ ❝♦♥t✐♥✉♦✉s str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t g(t) > h(t) ❛♥❞ t 0 h(s) g(s) ❞s < 1, 1 0 g(s) 2 ❞s = 1, 0 ≤ t < 1, t❤❡♥ ϕ 2 (t) ❞❡✜♥❡❞ ❜② ✭✶✳✷✹✮ ✐s str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ [0, 1)✳ ◆♦✇ ✇❡ ✇✐•• ✈❡r✐❢② t❤❛t ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ R 0 ✐s s❛t✐s✜❡❞✱ t❤❡♥ ϕ 2 (t) ✐s str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥✳ |❡♠✐♥❞ t❤❛t ϕ 2 (t) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ✿ ϕ 2 (t) = K(t) 2 + Φ 1 (t), 0 ≤ t < 1, ✇❤❡r❡ Φ 1 (t) = D(t) h(t) + N (t) (g(t) -h(t)) K(t), ✇✐t❤ D(t) = t 0 g(s) ❞s 1 + t 0 h(s) 2 ❞s - t 0 h(s) g(s) ❞s + t 0 h(s) ❞s t 0 g(s) 2 ❞s - t 0 h(s) g(s) ❞s , N (t) = C(t) t 0 h(s) ❞s + t 0 h(s) 2 ❞s t 0 g(s) ❞s ❛♥❞ K(t) = C(t) 2 + t 0 h(s) 2 ❞s 1 t g(s) 2 ❞s.
❍❡r❡

C(t) = 1 - t 0 h(s) g(s) ❞s. ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✐t ✐s s✉✣❝✐❡♥t t♦ ❝❤❡❝❦ t❤❛t Φ 1 (t) > 0 t♦ ♦❜t❛✐♥ ϕ 2 (t) > 0✳ |❡❝❛•• t❤❛t K(t) ✐s str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ ❞✉❡ t♦ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ t 0 h(s) g(s) ❞s < 1, ✭✶✳✹✶✮ ✇❡ ❤❛✈❡ C(t) > 0✳ ❈♦♥s❡q✉❡♥t•②✱ ✇❡ ♦❜t❛✐♥ N (t) > 0 ❜② t❤❡ ❝♦♥❞✐t✐♦♥s h(t) > 0 ❛♥❞ g(t) > 0. ✭✶✳✹✷✮ ❋✐♥❛••②✱ ✇❡ s❡❡ t❤❛t D(t) > 0 ❜② t❤❡ ❝♦♥❞✐t✐♦♥s 1 0 g(s) 2 ❞s = 1 ❛♥❞ ✭✶✳✹✶✮-✭✶✳✹✷✮✳ ❲❡ ❝♦♥❝•✉❞❡ t❤❛t ✇❡ ❤❛✈❡ ϕ 2 (t) > 0 ✐❢ ✇❡ s✉♣♣♦s❡ t❤❛t g(t) > h(t)✳ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✹✸ ✶✳✺ ❚❡st ❖✉r ♦❜•❡❝t✐✈❡ ✐s t♦ t❡st t❤❡ ❝♦♠♣♦s✐t❡ ♣❛r❛♠❡tr✐❝ ❤②♣♦t❤❡s✐s H 0 ❛♥❞ t♦ ❞♦ t❤✐s ✇❡ ✇✐•• ♣r♦♣♦s❡ ❛ st❛t✐st✐❝ ❜❛s❡❞ ♦♥ t❤❡ ▼❉❊ ϑ * ε ✳ |❡❝❛•• t❤❛t t❤❡ st❛rt✐♥❣ st❛t✐st✐❝ u ε (t) = X t -x t (ϑ * ε ) ε S (ϑ * ε , X t ) ✭✶✳✹✸✮ ❝♦♥✈❡r❣❡s t♦ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ u (t) = t 0 dW s S(ϑ, x s ) - T 0 T v S(ϑ, x s ) ẋs (ϑ) J(ϑ)S (ϑ, x v ) ❞s ❞W v t 0 Ṡ(ϑ, x s ) S(ϑ, x s ) ❞s. ❇② t❤❡ ■tô ❢♦r♠✉•❛✱ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ U t T = 1 √ T t 0 S (ϑ, x s ) du (s) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U t T = 1 √ T S(ϑ, x t ) u(t) - 1 √ T t 0 S ′ (ϑ, x s ) S(ϑ, x s ) u(s) ❞s ✇❤✐❝❤ •❡❛❞s t♦ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ U (ν) = W (ν) - 1 0 g (ϑ, r) dW (r) ν 0 h (ϑ, r) dr, 0 ≤ ν ≤ 1. ❚❤❡ •❛st st❡♣ ✐s t♦ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L [•] ❢r♦♠ ❚❤❡♦r❡♠ ✶✳✹✳✶ ❛♥❞ t♦ ♦❜t❛✐♥ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L [U ] (ν) = w ν . ◆♦✇ ✇❡ ❤❛✈❡ t♦ r❡❛•✐③❡ t❤❡ s✐♠✐✲ •❛r tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ t❤❡ ✏❡♠♣✐r✐❝❛•✑ ♣r♦❝❡ss u ε (•) ❣✐✈❡♥ ❜② ✭✶✳✹✸✮✱ ✐✳❡✳✱ ✇❡ ✭❢♦r♠❛••②✮ ❝❛•❝✉•❛t❡ U ε t T = 1 √ T t 0 S (ϑ * ε , X s ) du ε (s) . ✭✶✳✹✹✮ ❚❤❡♥ ✇❡ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L [•] t♦ t❤❡ ♣r♦❝❡ss U ε (•) ❛♥❞ ✇❡ s❤♦✇ t❤❛t t❤✐s st❛t✐st✐❝ ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss w ν , 0 ≤ ν ≤ 1✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st ψ ε = ✶ ■ {∆ε>cα} ✇✐t❤ ∆ ε = 1 T T 0 L [U ε ] (t) 2 dt =⇒ 1 0 w 2 ν dν ✇✐•• ❜❡ ❆❉❋ ✐♥ ✈✐❡✇ ♦❢ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ ∆ ε ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ S (•, •) ❛♥❞ ϑ✳ ✹✹ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ▲❡t ✉s r❡❛•✐③❡ t❤✐s ♣r♦❣r❛♠✳ ❲❡ ❤❛✈❡ ❢♦r t❤❡ ♣r♦❝❡ss ✭✶✳✹✹✮ t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U ε t T = 1 √ T S (ϑ * ε , X t ) u ε (t) - 1 √ T t 0 S ′ (ϑ * ε , X s ) S (ϑ * ε , X s ) u ε (s) ❞s. ✭✶✳✹✺✮ ■♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥s ĥ (ϑ, v) = 1 √ T J(ϑ) -1 Ṡ(ϑ, x v ), J(ϑ) = T 0 ẋs (ϑ) 2 ❞s ✭✶✳✹✻✮ ❛♥❞ ĝ (ϑ, v) = S(ϑ, x v ) -1 I (ϑ, v) , I (ϑ, v) = T v S(ϑ, x s ) ẋs (ϑ) ❞s ✭✶✳✹✼✮ ❛♥❞ t❤❡✐r ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑✱ r❡s♣❡❝t✐✈❡•② h ε (ϑ * ε , v) = 1 √ T J ε (ϑ * ε ) -1 Ṡ(ϑ * ε , X v ), g ε (ϑ * ε , v) = S(ϑ * ε , X v ) -1 I ε (ϑ * ε , v). ❍❡r❡ J ε (ϑ * ε ) = T 0 ẋs (ϑ * ε ) 2 ❞s, I ε (ϑ * ε , v) = T v S(ϑ * ε , X s ) ẋs (ϑ * ε ) ❞s. ◆♦t❡ t❤❛t ✐♥ t❤❡ ❢✉♥❝t✐♦♥s ĥ (•, •) ❛♥❞ ĝ (•, •) ✇❡ ♦♠✐t t❤❡ ♥♦r♠❛•✐③✐♥❣ ❝♦♥st❛♥ts ✐♥ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s h (•, •) ❛♥❞ g (•, •) ❞❡✜♥❡❞ ❜② ✭✶✳✶✾✮-✭✶✳✷✵✮✱ ❢♦r s✐♠♣•✐❝✐t② ♦❢ ❡①♣♦s✐t✐♦♥✱ ❜❡❝❛✉s❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ✉s❡❞ st❛t✐st✐❝ ✐s s✉❝❤ t❤❛t ✇❡ ❝❛♥ ❞♦ t❤✐s ✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝✳ ❚❤❡♥ ❞❡♥♦t❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ I 1,ε = C 1 (T ) s 0 g ε (ϑ * ε , v) 2 ❞v, I 2,ε = C 2 (T ) s 0 h ε (ϑ * ε , v) g ε (ϑ * ε , v) ❞v, I 3,ε = C 3 (T ) s 0 h ε (ϑ * ε , v) ❞v, I 4,ε = C 4 (T ) s 0 h ε (ϑ * ε , v) 2 ❞v ❛♥❞ I 5,ε = C 5 (T ) s 0 g ε (ϑ * ε , v) ❞v ♦❢ t❤❡ ✐♥t❡❣r❛•s Î1 = C 1 (T ) s 0 ĝ (ϑ, v) 2 ❞v, Î2 = C 2 (T ) s 0 ĥ (ϑ, v) ĝ (ϑ, v) ❞v, ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✹✺ Î3 = C 3 (T ) s 0 ĥ (ϑ, v) ❞v, Î4 = C 4 (T ) s 0 ĥ (ϑ, v) 2 ❞v ❛♥❞ Î5 = C 5 (T ) s 0 ĝ (ϑ, v) ❞v, ✇❤❡r❡ C 1 (T ) = 1 T 3 ✱ C 2 (T ) = √ T ✱ C 3 (T ) = T √ T ✱ C 4 (T ) = T 4 ❛♥❞ C 5 (T ) = 1 T 2 ✳ ❚❤✐s ❛••♦✇s ✉s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ ϕ 1,ε (•), ϕ 2,ε (•) ❛♥❞ ψ 2,ε (•) ♦❢ φ1 (•), φ2 (•) ❛♥❞ ψ2 (•) ❞❡✜♥❡❞ r❡s♣❡❝t✐✈❡•② ❜② ✭✶✳✷✸✮-✭✶✳✷✺✮✱ ✇❤❡r❡ ✇❡ r❡♣•❛❝❡ t❤❡ ❢✉♥❝t✐♦♥s g ❜② ĝ = 1 T ĝ (ϑ, s) ❛♥❞ h ❜② ĥ = T 2 √ T ĥ (ϑ, s)✱ ϕ 1,ε (s) = g ε -h ε + I 4,ε g ε -3I 2,ε g ε + I 5,ε h ε g ε + I 1,ε h ε + 2I 2,ε h ε + I 2 2,ε I 3,ε g 2 ε -2I 2,ε I 3,ε g 2 ε -I 5,ε h 2 ε + I 3,ε g 2 ε -I 2,ε I 4,ε g ε + 3I 2 2,ε g ε -2I 3,ε h ε g ε -2I 2,ε I 5,ε h ε g ε -I 3 2,ε g ε + I 2 2,ε I 5,ε h ε g ε + I 1,ε I 2 2,ε h ε -I 2 2,ε h ε + 2I 1,ε I 4,ε h ε -I 1,ε I 4,ε g ε + I 2 1,ε I 3,ε h 2 ε + I 1,ε I 4,ε I 5,ε h ε g ε + I 2,ε I 5,ε h 2 ε + I 1,ε I 5,ε h 2 ε + I 4,ε I 5,ε g 2 ε -I 1,ε I 2,ε I 5,ε h 2 ε -I 4,ε h ε -I 2,ε I 4,ε I 5,ε g 2 ε + 2I 1,ε I 3,ε h ε g ε -2I 1,ε I 2,ε I 3,ε h ε g ε -2I 1,ε I 3,ε h 2 ε -2I 1,ε I 2,ε h ε + I 1,ε I 2,ε I 4,ε g ε + 2I 2,ε I 3,ε h ε g ε -I 4,ε I 5,ε h ε g ε + I 3,ε h 2 ε -I 2 1,ε I 4,ε h ε , ϕ 2,ε (s) = 1 -2I 1,ε I 2 4,ε -3I 2,ε I 5,ε h ε + I 1,ε I 3,ε h ε + I 3,ε g ε -3I 2,ε I 3,ε g ε + I 4,ε I 5,ε g ε -I 3,ε h ε + 3I 2 2,ε I 5,ε h ε -2I 1,ε I 2,ε I 3,ε h ε -2I 1,ε I 2 2,ε I 4,ε -2I 2,ε I 4,ε I 5,ε g ε + 2I 2,ε I 3,ε h ε -I 3 2,ε I 5,ε h ε + I 1,ε I 2 2,ε I 3,ε h ε -2I 1,ε I 4,ε -I 3 2,ε I 3,ε g ε + I 2 4,ε + I 2 2,ε I 4,ε I 5,ε g ε + I 4,ε I 5,ε h ε + I 2 4,ε I 5,ε g ε -I 2,ε I 4,ε I 5,ε h ε + 2I 1,ε I 3,ε I 4,ε h ε + I 3,ε I 4,ε g ε + 3I 2 2,ε I 3,ε g ε + 2I 2 2,ε I 4,ε + 2I 4,ε -I 3,ε I 4,ε h ε + 6I 2 2,ε + I 1,ε I 2,ε I 4,ε I 5,ε h ε -I 2 1,ε I 3,ε I 4,ε h ε -I 1,ε I 3,ε I 4,ε g ε + I 5,ε h ε + I 1,ε I 2,ε I 3,ε I 4,ε g ε -I 1,ε I 2 4,ε I 5,ε g ε -I 1,ε I 4,ε I 5,ε h ε + 4I 1,ε I 2,ε I 4,ε -I 2,ε I 3,ε I 4,ε g ε -I 2 2,ε I 3,ε h ε + I 4 2,ε -4I 3 2,ε -4I 2,ε -4I 2,ε I 4,ε + I 2 1,ε I 2 4,ε ✭✶✳✹✽✮ ❛♥❞ ψ 2,ε (s) = h ε + I 3,ε h ε g ε + I 4,ε g ε + I 5,ε h 2 ε -3I 2,ε h ε -I 1,ε I 2,ε I 3,ε h 2 ε -2I 2,ε I 4,ε g ε + I 4,ε h ε -2I 2,ε I 5,ε h 2 ε + 3I 2 2,ε h ε + I 2 2,ε I 4,ε g ε ✹✻ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ + I 2 2,ε I 5,ε h 2 ε -I 3 2,ε h ε -I 3,ε I 4,ε h ε g ε + I 2 4,ε g ε -I 2,ε I 4,ε h ε + I 1,ε I 3,ε I 4,ε h ε g ε -I 1,ε I 2 4,ε g ε -I 1,ε I 4,ε h ε + I 1,ε I 2,ε I 4,ε h ε -I 3,ε h 2 ε + I 1,ε I 3,ε h 2 ε + I 3,ε I 4,ε g 2 ε + I 2,ε I 3,ε h 2 ε -2I 2,ε I 3,ε h ε g ε -2I 2,ε I 4,ε I 5,ε h ε g ε + 2I 4,ε I 5,ε h ε g ε -I 2,ε I 3,ε I 4,ε g 2 ε + I 2 2,ε I 3,ε h ε g ε + I 2 4,ε I 5,ε g 2 ε . ❍❡r❡ g ε = g ε (ϑ * ε , s) , h ε = h ε (ϑ * ε , s) ❛♥❞ I 1,ε ✱ I 2,ε ✱ I 3,ε ✱ I 4,ε ✱ I 5,ε ❛r❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ ♦❢ Î1 ✱ Î2 ✱ Î3 ✱ Î4 ✱ Î5 ✱ r❡s♣❡❝t✐✈❡•②✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t❡st ✇❡ ✐♥tr♦❞✉❝❡ ♦♥❡ ❝♦♥❞✐t✐♦♥ ❡•s❡✳ R 4 . ❲❡ s✉♣♣♦s❡ t❤❛t ϕ 2 (r) , r ∈ [0, 1) ❞❡✜♥❡❞ ❜② ✭✶✳✷✹✮ ✐s str✐❝t•② ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥✳ ❲❡ ❤❛✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② ✇✳r✳t✳ s ∈ [0, T ] (ε → 0) sup s∈[0,T ] |ϕ 2,ε (s) -φ2 (s)| -→ 0. ❚❤✐s ❝♦♥✈❡r❣❡♥❝❡ ✇❡ ♦❜t❛✐♥ ❞✉❡ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡st✐♠❛t♦r ❛♥❞ t❤❡ s♠♦♦t❤♥❡ss ♦❢ t❤❡ ❢✉♥❝t✐♦♥s g ε (•, •) ❛♥❞ h ε (•, •)✳ ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥ ϕ + 2,ε (s) =    ϕ 2,ε (s) -1 , ✐❢ ϕ 2,ε (s) > 0, 0, ❡•s❡, ✇❤✐❝❤ ❛s②♠♣t♦t✐❝❛••② ❝♦✐♥❝✐❞❡s ✇✐t❤ φ2 (s) -1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❝❤❛♥❣❡✳ ❍❡♥❝❡ ✇❡ ✜rst ❝♦♥s✐❞❡r ✭❢♦r♠❛••②✮ t❤❡ st❛t✐st✐❝ W ε (t) = U ε t T ✭✶✳✹✾✮ + 1 T t 0 s 0 ϕ + 2,ε (s)[λ 1 (T )ϕ 1,ε (s)h ε (ϑ * ε , q) + λ 2 (T )ψ 2,ε (s)g ε (ϑ * ε , q)]❞U ε q T ❞s, ✇❤❡r❡ λ 1 (T ) = T 2 √ T , λ 2 (T ) = 1 T ❛♥❞ U ε (•) ✐s ❞❡✜♥❡❞ ❜② ✭✶✳✹✺✮✳ ■❢ ✇❡ ♣r♦✈❡ t❤❛t W ε (t) -→ L [U ] (ν) = w ν , 0 ≤ ν ≤ 1 t❤❡♥ t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✇✐•• ❜❡ ❆❉❋✳ ❚❤❡ ♠❛✐♥ t❡❝❤♥✐❝❛• ♣r♦❜•❡♠ ✐♥ ❝❛rr②✐♥❣ ♦✉t t❤✐s ♣r♦❣r❛♠ ✐s t♦ ❞❡✜♥❡ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s K ε (ϑ * ε , s) = s 0 h ε (ϑ * ε , q) ❞U ε q T ✭✶✳✺✵✮ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✹✼ ❛♥❞ L ε (ϑ * ε , s) = s 0 g ε (ϑ * ε , q) ❞U ε q T . ✭✶✳✺✶✮ ❯♥❢♦rt✉♥❛t❡•② ✇❡ ❝❛♥ ♥♦t ❝❛•❝✉•❛t❡ t❤❡♠ ❛s t❤❡② ❛r❡ ✇r✐tt❡♥ ♥♦✇✱ ❜❡❝❛✉s❡ t❤❡ ✐♥t❡❣r❛♥❞ ❝♦♥t❛✐♥s t❤❡ ▼❉❊ ϑ * ε ❛♥❞ t❤✐s ❡st✐♠❛t♦r ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❤♦•❡ tr❛✲ •❡❝t♦r② X ε = (X t , 0 ≤ t ≤ T )✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s K ε (ϑ * ε , s) ❛♥❞ L ε (ϑ * ε , s) ❛r❡ ♥♦t ✇❡•• ❞❡✜♥❡❞✳ ❚♦ ❛✈♦✐❞ t❤✐s ♣r♦❜•❡♠ ✇❡ ✉s❡ ❛♥ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ ❛♣♣•✐❝❛t✐♦♥ ♦❢ t❤❡ ■tô ❢♦r♠✉•❛✱ ✐✳❡✳✱ ✇❡ r❡♣•❛❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ❜② t❤❡ ♦r❞✐♥❛r② ♦♥❡s✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ✇❛s ❛♣♣•✐❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✐♥ ❬✹✹(✳ ■♥tr♦❞✉❝❡ t❤❡ st❛t✐st✐❝ K (ϑ, s) = s 0 ĥ (ϑ, q) ❞U q T = 1 √ T s 0 ĥ (ϑ, q) S (ϑ, x q ) ❞u(q), ✭✶✳✺✷✮ ✇❤❡r❡ t❤❡ ♣r♦❝❡ss U (•) ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ĥ (•, •) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✶✳✶✻✮ ❛♥❞ ✭✶✳✹✻✮✱ r❡s♣❡❝t✐✈❡•②✳ ■♥❞❡❡❞ t❤❡ ■tô ❢♦r♠✉•❛ ❣✐✈❡s ✉s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥
❞ ĥ(ϑ, q)S(ϑ, x q )u(q) = ĥ′ (ϑ, q)S(ϑ, x q ) + ĥ(ϑ, q)S ′ (ϑ, x q )S(ϑ, x q ) u(q) ❞q + ĥ (ϑ, q) S (ϑ, x q ) ❞u(q). ❍❡r❡ ĥ′ (ϑ, q) ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ĥ (ϑ, q) ✇✳r✳t✳ q✱ ❣✐✈❡♥ ❜② t❤❡ ❢♦

••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ĥ′ (ϑ, q) = 1 √ T J(ϑ) -1 Ṡ′ (ϑ, x q (ϑ)) S (ϑ, x q (ϑ)) . ✭✶✳✺✸✮ ❚❤❡r❡❢♦r❡ t❤❡ st❛t✐st✐❝ K (ϑ, •) ❞❡✜♥❡❞ ❜② ✭✶✳✺✷✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ K (ϑ, s) = 1 √ T ĥ (ϑ, s) S (ϑ, x s ) u(s) - 1 √ T s 0
ĥ′ (ϑ, q) S (ϑ, x q ) + ĥ (ϑ, q) S ′ (ϑ, x q ) S (ϑ, x q ) u(q) ❞q.

❍❡♥❝❡ ✇❡ ♦❜t❛✐♥ ❢♦r t❤❡ ♣r♦❝❡ss ✭✶✳✺✵✮ t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ K ε (ϑ * ε , s) = 1 √ T h ε (ϑ * ε , s) S (ϑ * ε , X s ) u ε (s) - 1 √ T s 0 (h ′ ε (ϑ * ε , q)S(ϑ * ε , X q ) + h ε (ϑ * ε , q)S ′ (ϑ * ε , X q )S(ϑ * ε , X q )) u ε (q) ❞q ✭✶✳✺✹✮ ✹✽ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❛♥❞ t❤❡ ✐♥t❡❣r❛• ✐s ♥♦✇ ✇❡•• ❞❡✜♥❡❞✳ ❍❡r❡ h ′ ε (ϑ * ε , q) = 1 √ T J ε (ϑ * ε ) -1 Ṡ′ (ϑ * ε , X q ) S (ϑ * ε , X q ) . ❙✐♠✐•❛r•②✱ t❤❡ ♣r♦❝❡ss ✭✶✳✺✶✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ L ε (ϑ * ε , s) = 1 √ T g ε (ϑ * ε , s) S (ϑ * ε , X s ) u ε (s) - 1 √ T s 0 (g ′ ε (ϑ * ε , q)S(ϑ * ε , X q ) + g ε (ϑ * ε , q)S ′ (ϑ * ε , X q )S(ϑ * ε , X q )) u ε (q) ❞q, ✭✶✳✺✺✮ ✇❤❡r❡ g ′ ε (ϑ * ε , q) = -S(ϑ * ε , X q ) -1 S ′ (ϑ * ε , X q ) T q S(ϑ * ε , X s ) ẋs (ϑ * ε ) ❞s -ẋq (ϑ * ε ) ✐s t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ ĝ′ (ϑ, q) = -S(ϑ, x q (ϑ)) -1 S ′ (ϑ, x q (ϑ)) T q S(ϑ, x s (ϑ)) ẋs (ϑ) ❞s -ẋq (ϑ). ✭✶✳✺✻✮ ❚❤❡♥ t❤❡ ❢♦r♠❛• ❡①♣r❡ss✐♦♥ ✭✶✳✹✾✮ ❢♦r W ε (t) ❝❛♥ ❜❡ r❡♣•❛❝❡❞ ❜② Wε (t) = U ε t T + 1 T t 0 ϕ + 2,ε (s) [λ 1 (T ) ϕ 1,ε (s) K ε (ϑ * ε , s) + λ 2 (T ) ψ 2,ε (s) L ε (ϑ * ε , s)] ❞s, ✭✶✳✺✼✮ ✇❤❡r❡ λ 1 (T ) = T 2 √ T , λ 2 (T ) = 1 T ❛♥❞ t❤❡ ♣r♦❝❡ss❡s U ε (•)✱ K ε (ϑ * ε , •) ❛♥❞ L ε (ϑ * ε , •) ❛❞♠✐t t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ✭✶✳✹✺✮✱ ✭✶✳✺✹✮ ❛♥❞ ✭✶✳✺✺✮✱ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ t❡st ✐s ❣✐✈❡♥ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❚❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✶✳✺✳✷✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ r❡❣✉•❛r✐t② R ❛♥❞ R 4 ❛r❡ ❢✉•✜••❡❞✱ t❤❡♥ t❤❡ t❡st ψ ε = ✶■ {∆ε>cα} , P ∆ > c α = α, ✇✐t❤ ∆ ε = 1 T T 0 Wε (t) 2 ❞t, ∆ ≡ 1 0 w 2 ν ❞ν ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✳ Pr♦♦❢✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s H 0 ✱ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∆ ε =⇒ ∆ ✭✶✳✺✽✮ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✹✾ ❤♦•❞s✳ |❡❝❛•• t❤❛t t❤❡ ♣r♦❝❡ss U ε (•) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U ε t T = 1 √ T S (ϑ * ε , X t ) u ε (t) - 1 √ T t 0 S ′ (ϑ * ε , X s ) S (ϑ * ε , X s ) u ε (s) ❞s. ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ ❛•r❡❛❞② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② ✉♥✐❢♦r♠•② ✇✳r✳t✳ s ∈ [0, T ] ✭❛s ε → 0✮ sup s∈[0,T ] |X s -x s (ϑ)| -→ 0, sup s∈[0,T ] |u ε (s) -u (s)| -→ 0. ❋✉rt❤❡r✱ ✇❡ ❝❛♥ ✇r✐t❡ |S (ϑ * ε , X s ) -S (ϑ, x s )| ≤ |S (ϑ * ε , X s ) -S (ϑ, X s )| + |S (ϑ, X s ) -S (ϑ, x s )| ≤ |ϑ * ε -ϑ| | Ṡ( θ, X s )| + |X s -x s | |S ′ (ϑ, Xs )|. ❍❡r❡ | θ -ϑ| ≤ |ϑ * ε -ϑ| ❛♥❞ | Xs -X s | ≤ |x s (ϑ * ε ) -X s | ≤ |x s (ϑ * ε ) -x s (ϑ)| + |x s (ϑ) -X s | → 0. ❚❤✐s ❝♦♥✈❡r❣❡♥❝❡ ✐s ✉♥✐❢♦r♠ ✇✳r✳t✳ s ∈ [0, T ]✳ ❚❤❡♥ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s Ṡ(ϑ, x) ❛♥❞ S ′ (ϑ, x) ❛r❡ ❜♦✉♥❞❡❞ ❜② t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✱ t❤❡ ♣r♦❝❡ss X s ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠•② ✇✳r✳t✳ s ∈ [0, T ] t♦ x s (ϑ) ❛♥❞ ❞✉❡ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡st✐♠❛t♦r ϑ * ε ✱ ✇❡ ♦❜t❛✐♥ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ s ∈ [0, T ] ✭✐♥ ♣r♦❜❛❜✐•✐t②✮ sup s∈[0,T ] |S (ϑ * ε , X s ) -S (ϑ, x s )| -→ 0. ▼♦r❡♦✈❡r✱ s✐♠✐•❛r ❛r❣✉♠❡♥ts ❣✐✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ s ∈ [0, T ] ✭✐♥ ♣r♦❜❛❜✐•✐t②✮ sup s∈[0,T ] |S ′ (ϑ * ε , X s ) -S ′ (ϑ, x s )| -→ 0. ❚❤❡r❡❢♦r❡✱ ✇❡ ♦❜t❛✐♥ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ t ∈ [0, T ] ✭✐♥ ♣r♦❜❛❜✐•✐t②✮ U ε t T -→ 1 √ T S(ϑ, x t ) u(t) - 1 √ T t 0 S ′ (ϑ, x s ) S(ϑ, x s ) u(s) ❞s = 1 √ T t 0 S(ϑ, x s ) ❞u(s) = U t T . ✺✵ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ◆♦✇ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t K ε (ϑ * ε , s) -→ K(ϑ, s)✱ ✇❤❡r❡ K ε (ϑ * ε , s) = 1 √ T h ε (ϑ * ε , s) S (ϑ * ε , X s ) u ε (s) - 1 √ T s 0 (h ′ ε (ϑ * ε , q)S(ϑ * ε , X q ) + h ε (ϑ * ε , q)S ′ (ϑ * ε , X q )S(ϑ * ε , X q )) u ε (q) ❞q ❛♥❞ K (ϑ, s) = 1 √ T ĥ (ϑ, s) S (ϑ, x s ) u(s) - 1 √ T s 0
ĥ′ (ϑ, q) S (ϑ, x q ) + ĥ (ϑ, q) S ′ (ϑ, x q ) S (ϑ, x q ) u(q) ❞q,

✇❤❡r❡ ĥ (•, •) ❛♥❞ ĥ′ (•, •) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✶✳✹✻✮ ❛♥❞ ✭✶✳✺✸✮✱ r❡s♣❡❝t✐✈❡•②✳ ❲❡ ❤❛✈❡ Ṡ (ϑ * ε , X s ) -Ṡ (ϑ, x s ) ≤ Ṡ (ϑ * ε , X s ) -Ṡ (ϑ, X s ) + Ṡ (ϑ, X s ) -Ṡ (ϑ, x s ) ≤ |ϑ * ε -ϑ| | S( θ, X s )| + |X s -x s | | Ṡ′ (ϑ, Xs )| ≤ C1 |ϑ * ε -ϑ| + C2 |X s -x s | . ❍❡r❡ C1 ❛♥❞ C2 ❛r❡ s♦♠❡ ❝♦♥st❛♥ts✳ ❉✉❡ t♦ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✱ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ S(ϑ, x) ✇✳r✳t✳ ϑ ❛♥❞ Ṡ′ (ϑ, x) ❛r❡ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s✳ ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② sup s∈[0,T ] Ṡ (ϑ * ε , X s ) -Ṡ (ϑ, x s ) -→ 0. ❋✉rt❤❡r✱ ✇❡ ❤❛✈❡ ✭❛s ε → 0✮ J ε (ϑ * ε ) -J(ϑ) = T 0 ẋs (ϑ * ε ) 2 ❞s - T 0 ẋs (ϑ) 2 ❞s -→ 0. ❍❡♥❝❡ ✇❡ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② sup s∈[0,T ] h ε (ϑ * ε , s) -ĥ (ϑ, s) -→ 0. ■♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✱ ✐t ✐s s❤♦✇♥ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② sup s∈[0,T ] h ′ ε (ϑ * ε , s) -ĥ′ (ϑ, s) -→ 0. ❚❤❡r❡❢♦r❡✱ ❜② t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ u ε (s) t♦ u(s) ✇✳r✳t✳ s ∈ [0, T ]✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② K ε (ϑ * ε , s) -→ K(ϑ, s). ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✺✶ ◆♦✇ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t L ε (ϑ * ε , s) -→ L(ϑ, s), ✇❤❡r❡ L ε (ϑ * ε , s) = 1 √ T g ε (ϑ * ε , s) S (ϑ * ε , X s ) u ε (s) - 1 √ T s 0 (g ′ ε (ϑ * ε , q)S(ϑ * ε , X q ) + g ε (ϑ * ε , q)S ′ (ϑ * ε , X q )S(ϑ * ε , X q )) u ε (q) ❞q ❛♥❞ L (ϑ, s) = 1 √ T ĝ (ϑ, s) S (ϑ, x s ) u(s) - 1 √ T s 0 (ĝ ′ (ϑ, q) S (ϑ, x q ) + ĝ (ϑ, q) S ′ (ϑ, x q ) S (ϑ, x q )) u(q) ❞q, ✇❤❡r❡ ĝ (•, •) ❛♥❞ ĝ′ (•, •) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✶✳✹✼✮ ❛♥❞ ✭✶✳✺✻✮✱ r❡s♣❡❝t✐✈❡•②✳ ❖❜s❡r✈❡ t❤❛t ❢♦r s ∈ [0, T ]✱ ✇❡ ❤❛✈❡ |I ε (ϑ * ε , s) -I(ϑ, s)| ≤ T s |S(ϑ * ε , X v ) ( ẋv (ϑ * ε ) -ẋv (ϑ))| ❞v + T s | ẋv (ϑ) (S(ϑ * ε , X v ) -S(ϑ, x v (ϑ)))| ❞v ≤ T s |S(ϑ * ε , X v )| | ẋv (ϑ * ε ) -ẋv (ϑ)| ❞v + |ϑ * ε -ϑ| T s | ẋv (ϑ)| | Ṡ( θ, X v )|❞v + T s | ẋv (ϑ)| |X v -x v (ϑ)| |S ′ (ϑ, Xv )|❞v ≤ C3 |ϑ * ε -ϑ| + C4 |ϑ * ε -ϑ| + C5 sup s∈[0,T ] |X s -x s (ϑ)| (T -s) , ✇❤❡r❡ C3 ✱ C4 ❛♥❞ C5 ❛r❡ ❝♦♥st❛♥ts✳ ❚❤❡ ❡st✐♠❛t♦r ϑ * ε ✐s ❝♦♥s✐st❡♥t✱ t❤❡ ♣r♦❝❡ss X s ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠•② ✇✳r✳t✳ s ∈ [0, T ] t♦ x s (ϑ) ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❜♦✉♥❞❡❞ ❞✉❡ t♦ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ r❡❣✉•❛r✐t② R✳ ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② ✉♥✐❢♦r♠•② ✇✳r✳t✳ s ∈ [0, T ] sup s∈[0,T ] |I ε (ϑ * ε , s) -I(ϑ, s)| -→ 0. ▼♦r❡♦✈❡r✱ ✇❡ ♣r♦✈❡❞ ❛•r❡❛❞② t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ s ∈ [0, T ] ♦❢ u ε (s) t♦ u(s) ❛♥❞ S(ϑ * ε , X s ) t♦ S(ϑ, x s )✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② sup s∈[0,T ] |g ε (ϑ * ε , s) -ĝ (ϑ, s)| -→ 0. ✺✷ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❙✐♠✐•❛r•②✱ ❜② t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✱ ✐t ✐s s❤♦✇♥ t❤❛t ✭✐♥ ♣r♦❜❛❜✐•✐t②✮ sup s∈[0,T ] |g ′ ε (ϑ * ε , s) -ĝ′ (ϑ, s)| -→ 0. ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② L ε (ϑ * ε , s) -→ L (ϑ, s) . ❉✉❡ t♦ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ❛♥❞ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡st✐♠❛t♦r ϑ * ε ✱ s✐♠✐•❛r ❛r❣✉♠❡♥ts ❣✐✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐•✐t② ✉♥✐❢♦r♠•② ✇✳r✳t✳ s ∈ [0, T ] sup s∈[0,T ] ψ 2,ε (s) -ψ2 (s) -→ 0 ❛♥❞ sup s∈[0,T ] |ϕ 1,ε (s) -φ1 (s)| -→ 0, sup s∈[0,T ] |ϕ 2,ε (s) -φ2 (s)| -→ 0. ❋✐♥❛••②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♠❡♥t✐♦♥❡❞ ✐♥ ✭✶✳✺✽✮ ✐s ♣r♦✈❡❞ ❛♥❞ ✉s✐♥❣ t❤✐s r❡s✉•t t❤❡ t❡st ψ ε ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✱ ✇❤✐❝❤ ❝♦♠♣•❡t❡s t❤❡ ♣r♦♦
♦❜t❛✐♥ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ♠❡♥t✐♦♥❡❞ ✐♥ ❬✸✹(✱ ✇❡ ♣✉t h(ϑ, r) = g(ϑ, r) ✐♥ ✭✶✳✷✷✮ ❛♥❞ ✇❡ ♦❜t❛✐♥ I 1 = I 2 = I 4 = r 0 h(ϑ, q) 2 ❞q, I 3 = I 5 = r 0 h(ϑ, q) ❞q. ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✇r✐t❡ ϕ 1 (r) = h -h + I 1 h -3I 1 h + I 3 h 2 + I 3 h 2 + 2I 1 h -I 2 1 h -2I 1 I 3 h 2 -I 2 1 I 3 h 2 -I 1 h + I 2 1 I 3 h 2 -I 2 1 h + 3I 2 1 h + I 1 I 3 h 2 -2I 1 I 3 h 2 -2I 2 1 h + 2I 1 I 3 h 2 + I 2 1 I 3 h 2 + I 3 1 h + I 3 h 2 + I 1 h + 2I 2 1 h -I 2 1 h + I 3 1 h + I 2 1 I 3 h 2 -2I 2 1 I 3 h 2 -I 3 1 h + I 1 I 3 h 2 + I 1 I 3 h 2 + 2I 1 I 3 h 2 -I 3 1 h -I 2 1 I 3 h 2 -I 3 h 2 -2I 1 I 3 h 2 + I 2 1 I 3 h 2 -I 1 I 3 h 2 -2I 3 h 2 ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ✺✸ = 0, ϕ 2 (r) = 1 + I 3 h -3I 1 I 3 h + I 1 I 3 h + I 3 h -3I 1 I 3 h + I 1 I 3 h -I 3 h + 6I 2 1 -2I 4 1 + 3I 2 1 I 3 h -2I 2 1 I 3 h + 2I 1 I 3 h -I 3 1 I 3 h + I 3 1 I 3 h -I 3 1 I 3 h + I 2 1 -4I 1 + I 1 I 3 h -I 2 1 I 3 h + 2I 2 1 I 3 h + I 1 I 3 h -I 2 1 I 3 h + 2I 3 1 + I 2 1 I 3 h -4I 3 1 + I 3 1 I 3 h -I 3 1 I 3 h -I 2 1 I 3 h + I 3 1 I 3 h -I 2 1 I 3 h + 4I 3 1 + 3I 2 1 I 3 h -I 3 1 I 3 h + I 4 1 -2I 3 1 + I 4 1 + 2I 1 -2I 2 1 + I 3 1 I 3 h -2I 2 1 I 3 h -I 1 I 3 h -I 2 1 I 3 h -4I 2 1 = 1 + I 3 h -I 1 I 3 h + I 2 1 -2I 1 = (1 -I 1 )(1 + I 3 h -I 1 ) ❛♥❞ ψ 2 (r) = h + I 3 h 2 + I 1 h + I 3 h 2 -3I 1 h -2I 1 I 3 h 2 -2I 2 1 h -2I 1 I 3 h 2 -I 3 1 h + I 3 1 h + I 2 1 I 3 h 2 -I 3 1 h -I 1 I 3 h 2 + I 2 1 h + I 1 h -I 2 1 h -I 3 h 2 + 3I 2 1 h -I 2 1 h + I 3 1 h + I 1 I 3 h 2 + I 1 I 3 h 2 + I 1 I 3 h 2 -I 2 1 I 3 h 2 + 2I 2 1 I 3 h 2 -I 2 1 I 3 h 2 -2I 2 1 I 3 h 2 + 2I 1 I 3 h 2 + I 2 1 I 3 h 2 = h (1 + I 3 h -I 1 ), ✇❤❡r❡ h = h(ϑ, r)✳ ❍❡♥❝❡ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✷✻✮ ✇✐•• ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ ✿ L[U ](ν) = U (ν) + ν 0 r 0 h(r) N(r) -1 h(q) ❞U (q) ❞r = w ν , ✭✶✳✺✾✮ ✇❤❡r❡ N(r) = 1 r h(q) 2 ❞q ❛♥❞ U (ν) = W (ν) - 1 0 h(r) ❞W (r) ν 0 h(r) ❞r, ✇✐t❤ W (ν) ❛♥❞ w ν , 0 ≤ ν ≤ 1 ❛r❡ s♦♠❡ st❛♥❞❛r❞ ❲✐❡♥❡r ♣r♦❝❡ss❡s✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L [•] ❣✐✈❡♥ ❜② ✭✶✳✺✾✮ ♦❢ t❤❡ •✐♠✐t ♣r♦❝❡ss U (•) ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♦♥❡ ♣r♦♣♦s❡❞ ❜② ❑❤♠❛•❛❞③❡ ❬✸✵(✳ ✺✹ ❈❍❆P❚❊| ✶✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| P❊|❚❯|❇❊❉ ❉(◆❆▼■❈❆▲ ❙(❙❚❊▼❙ ❈❤❛♣t❡r ✷ ❖♥ •♦❋ ❚❡st ❢♦r ■♥❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ Pr♦❝❡ss ✷✳✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤✐s ❝❤❛♣t❡r ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t✇♦ ❆❉❋ •♦❋ t❡sts ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛• ❛♥❞ s❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝s✳ ❙✉♣♣♦s❡ t❤❛t ✇❡ ♦❜s❡r✈❡ ❛ ♣❡r✐♦❞✐❝ ✐♥❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss X n = (X t , 0 ≤ t ≤ T = nτ ) ✇✐t❤ ❦♥♦✇♥ ♣❡r✐♦❞ τ > 0✳ ❚❤❡ ♠❡❛♥ Λ (t) ❛♥❞ ✐♥t❡♥s✐t② ❢✉♥❝t✐♦♥s λ (t) s❛t✐s❢② t❤❡ r❡•❛t✐♦♥s EX t = Λ (t) , Λ (t) = t 0 λ (s) ❞s. ❖✉r ❛✐♠ ✐s t♦ ❝♦♥str✉❝t t✇♦ ❆❉❋ •♦❋ t❡sts ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 : Λ(•) ∈ {Λ(ϑ, •), ϑ ∈ Θ = (a, b)} , ✭✷✳✶✮ ✇❤❡r❡ a > 0 ❛♥❞ b < ∞✳ ❚❤❡s❡ t❡sts ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛✲ t✐♦♥ L [•] ✭✶✳✷✻✮ ❛♣♣•✐❡❞ t♦ t❤❡ ❡♠♣✐r✐❝❛• ❛♥❞ s❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝s✳ ■♥ t❤❡ •✐t❡r❛t✉r❡ ♦♥ t❤❡ st❛t✐st✐❝❛• ❛♥❛•②s✐s ♦❢ ♣♦✐♥t ♣r♦❝❡ss❡s ♥✉♠❡r♦✉s •♦❋ t❡sts ❢♦r ❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ✇❡r❡ ♣r♦♣♦s❡❞✱ ❡✳❣✳✱ ✐♥ ❬✹✼(✱ ❬✶✹( ❛♥❞ ❬✶(✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✐♥ ❬✶( ❛ t❡st ♣r♦❝❡❞✉r❡ ✇❛s ✐♥tr♦❞✉❝❡❞ ❢♦r t❡st✐♥❣ ✇❡t❤❡r ❛ ♣♦✐♥t ♣r♦❝❡ss ✐s ❤♦♠♦❣❡♥❡♦✉s ✿ ❋✐rst ✇❤❡t❤❡r t❤❡ ♣r♦❝❡ss ✐s ❛ ♠✐①❡❞ P♦✐ss♦♥ ♣r♦❝❡ss ❛♥❞ t❤❡♥ ✇❤❡t❤❡r t❤❡ ♣r♦❝❡ss ✐s ❛ r❡♥❡✇❛• ♣r♦❝❡ss✳ ❋✉rt❤❡r✱ ✐♥ ❬✺(✱ t❤❡ ❛✉t❤♦rs ❞❡✈❡•♦♣❡❞ ❛ t❡st ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛• ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐s ❝♦♥s✐st❡♥t✳ ❆♥♦t❤❡r •♦❋ t❡st ✇❛s ❣✐✈❡♥ ✐♥ ❬✺✾( ❢♦r ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s ❜❛s❡❞ ♦♥ ✺✺ ✺✻ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❜② ♠❡❛♥ ❞✐st❛♥❝❡s✳ ▼♦r❡♦✈❡r✱ •♦❋ t❡sts ❜❛s❡❞ ♦♥ t❤❡ ❈r❛♠ér✲ ✈♦♥ ▼✐s❡s st❛t✐st✐❝s ✇❡r❡ ♣r♦♣♦s❡❞ ✐♥ ❬✺✽( ❢♦r t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥✳ P♦✇❡r ❝♦♠♣❛r✐s♦♥s s❤♦✇❡❞ t❤❛t t❤❡s❡ st❛t✐st✐❝s ❣✐✈❡ ❣♦♦❞ ♦✈❡r❛•• t❡sts ♦❢ ✜t✳ ❖t❤❡r ❝♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥ t❤❡ P❡❛rs♦♥✱ s♠♦♦t❤ t❡sts ❛♥❞ ❛ ♠♦❞✐✜❡❞ ❑♦•♠♦❣♦r♦✈✲ ❙♠✐r♥♦✈ t❡st ❛r❡ ❣✐✈❡♥ ✐♥ ❬✹(✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ❛♥❞ ♠♦r❡ ❣❡♥❡r❛• ♣♦✐♥t ♣r♦✲ ❝❡ss❡s✱ t❤❡ •♦❋ t❡sts ❢♦r ♣❛r❛♠❡tr✐❝ ❤②♣♦t❤❡s✐s ✇❡r❡ st✉❞✐❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✱ ❡✳❣✳✱ ❬✷✺(✱ ❬✷(✱ ❬✹✾(✱ ❬✶✶( ❛♥❞ ❬✹✷(-❬✹✸(✳ ◆♦t❡ t❤❛t t❤❡ ❡st✐♠❛t♦r ✉s❡❞ ✐♥ ❛•• t❤❡s❡ ✇♦r❦s ✐s ❛•✇❛②s t❤❡ ▼▲❊ ❛♥❞ t❤❡ ♣r♦♣♦s❡❞ t❡sts ❛r❡ ❆❉❋✳ ■♥ ❬✹✷(✱ t❤❡ ❛✉✲ t❤♦r ♣r♦♣♦s❡❞ t✇♦ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ •✐♠✐t st❛t✐st✐❝s✱ ✇❤✐❝❤ ❛••♦✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❆❉❋ t❡sts✳ ❍♦✇❡✈❡r✱ ✐♥ ❬✹✸(✱ ✐t ✇❛s s❤♦✇♥ t❤❛t ❛ s♣❡❝✐❛• ❝❤❛♥❣❡ ♦❢ t✐♠❡ tr❛♥s❢♦r♠❡❞ t❤❡ •✐♠✐t s❝♦r❡✲❢✉♥❝t✐♦♥ ♣r♦❝❡ss❡s ✐♥t♦ t❤❡ ❇r♦✇♥✐❛♥ ❜r✐❞❣❡ ❛♥❞ t❤❡ ❆❉❋ t❡sts ✇❡r❡ ❝♦♥str✉❝t❡❞ ❞✉❡ t♦ t❤✐s ♣r♦♣❡rt②✳ ❚❤❡ ❝❛s❡ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ s❤✐❢t ♣❛r❛♠❡t❡r ✭♦♥❡ ❞✐♠❡♥s✐♦♥❛• ❝❛s❡✮ ✇❛s st✉❞✐❡❞ ✐♥ ❬✶✶(✳ ❚❤✐s ✇♦r❦ s❤♦✇❡❞ t❤❛t t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s st❛t✐st✐❝ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ✭s❤✐❢t✮ ♣❛r❛♠❡t❡r ❛♥❞ t❤❡r❡✲ ❢♦r❡ t❤❡ st❛t✐st✐❝ ✐s ❛s②♠♣t♦t✐❝❛••② ♣❛r❛♠❡t❡r ❢r❡❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ s✐♠♣•❡ ♥✉•• ❤②♣♦t❤❡s✐s✱ s❡✈❡r❛• r❡s✉•ts ❝♦♥❝❡r♥✐♥❣ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s ❛♥❞ ❑♦•♠♦❣♦r♦✈✲❙♠✐r♥♦✈ t②♣❡ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❢♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ ♣r♦❝❡ss❡s ✇❡r❡ ♣r❡s❡♥t❡❞ ✐♥ ❬✶✷(✳ ❆s ♠♦❞❡•s ♦❢ ♦❜s❡r✈❛t✐♦♥s✱ t❤❡ ❛✉t❤♦rs t❛❦❡ ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✇✐t❤ s♠❛•• ♥♦✐s❡✱ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✱ P♦✐ss♦♥ ♣r♦❝❡ss ❛♥❞ s❡•❢✲❡①❝✐t✐♥❣ ♣♦✐♥t ♣r♦❝❡ss❡s✳ ▲❡t ✉s r❡✈✐❡✇ s♦♠❡ ♦❢ t❤❡ r❡•❛t❡❞ r❡s✉•ts ❢♦r t❤❡ s✐♠♣•❡ ♥✉•• ❤②♣♦t❤❡s✐s✱ ✐✳❡✳✱ Θ = {ϑ 0 } ✭s❡❡✱ ❡✳❣✳✱ ❬✹✺( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮ H 0 : Λ (t) = Λ (ϑ 0 , t) .
■♥tr♦❞✉❝❡ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ .

η * n = n Λ (ϑ 0 , τ ) 2 τ 0 Λn (r) -Λ (ϑ 0 , r) 2 λ (ϑ 0 , r) ❞r. ❍❡r❡ Λn (•) ✐s t❤❡ ❡♠♣✐r✐❝❛• ♠❡❛♥ ❢✉♥❝t✐♦♥ ♦♥ ♦♥❡ ♣❡r✐♦❞ ❝♦♥str✉❝t❡❞ ❛s ❢♦••♦✇s ✿ Λn (r) = 1 n n j=1 X τ (j-1)+r -X τ (j-1) = 1 n n j=1 X j (r), 0 ≤ r ≤ τ. ✭✷✳✷✮ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✺✼ ❚❤❡ ❢♦••♦✇✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ ✭✉♥❞❡r H 0 ✮ η * n =⇒ ∆ = 1 0 w 2 ν ❞ν ❤♦•❞s✱ ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ■t ✐s ❝•❡❛r t❤❛t t❤❡ •✐♠✐t st❛t✐st✐❝ ∆ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♠♦❞❡• Λ (•, •)✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t❡st ϕ * n = ✶ ■ {η * n >cα} ✇✐t❤ P ∆ > c α = α, α ∈ (0, 1) ✐s ❆❉❋✳ ▲❡t ✉s r❡❝❛•• ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s ✭✷✳✶✮✳ ❚❤❡ ❞❡t❛✐•s ❝♦♥❝❡r♥✐♥❣ t❤✐s r❡s✉•t ❝❛♥ ❜❡ ❢♦✉♥❞✱ ❡✳❣✳✱ ✐♥ ❬✹✷(-❬✹✺(✳ ■♥tr♦❞✉❝❡ t❤❡ ▼▲❊ θn ♦❢ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ ❝♦♥str✉❝t❡❞ ❜② t❤❡ ♦❜✲ s❡r✈❛t✐♦♥s ♦❢ ❛ ♣❡r✐♦❞✐❝ P♦✐ss♦♥ ♣r♦❝❡ss X n ✳ ❲❡ s✉♣♣♦s❡ t❤❛t θn ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• ✭❛s n → +∞✮ ✉♥❞❡r r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s ✭s❡❡ ❬✸✽( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮ √ n θn -ϑ =⇒ N 0, I(ϑ) -1 , I(ϑ) = τ 0 λ (ϑ, s) 2 λ (ϑ, s)
✭✷✳✹✮ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❢♦••♦✇s ✐♠♠❡❞✐❛t❡•② ❢r♦♠ t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ √ n Λn (r) -Λ(ϑ, r) =⇒ W (Λ(ϑ, r)) ∼ N (0, Λ(ϑ, r)) , ✇❤❡r❡ W (•) ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✱ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s ✭✷✳✶✮✱ ✭s❡❡✱ ❡✳❣✳✱ ❬✹✷( ❢♦r ❞❡t❛✐•s✮ Ṽn =⇒ 1 0 Ũ (ν) 2 ❞ν ≡ Ṽ , ✇❤❡r❡ Ũ (ν) = w(ν) - 1 0 h (ϑ, s) ❞w(s) ν 0 h (ϑ, s) ❞s, 0 ≤ ν ≤ 1 ✭✷✳✺✮ ✺✽ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❛♥❞ h (ϑ, s) = λ (ϑ, v(s)) λ (ϑ, v(s)) Λ(ϑ, τ ) I(ϑ) , 1 0 h (ϑ, s) 2 ❞s = 1. ❍❡r❡ w(ν), 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ❛♥❞ v(s) ✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ s = Λ(ϑ, v)Λ(ϑ, τ ) -1 ✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ •✐♠✐t ♣r♦❝❡ss Ṽ ❞❡♣❡♥❞s str♦♥❣•② ♦♥ t❤❡ ♠♦❞❡• Λ(•, •) ❛♥❞ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❍❡♥❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡st φn = ✶ ■ { Ṽn>cα} ✐s ♥♦t ❆❉❋ ❛♥❞ t❤❡ t❤r❡s❤♦•❞ c α ❞❡✜♥❡❞ ❛s s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ P Ṽ > c α = α, α ∈ (0, 1) ❞❡♣❡♥❞s ♦♥ Λ(•, •) ✭❦♥♦✇♥✮ ❛♥❞ ϑ ✭✉♥❦♥♦✇♥✮✱ ✐✳❡✳✱ c α = c α (Λ, ϑ)✳ ❖♥❡ ✇❛② t♦ ✜♥❞ t❤❡ t❤r❡s❤♦•❞ ✐s ✉s✐♥❣ ♥✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s t♦ ❝❛•❝✉•❛t❡ t❤❡ ❢✉♥❝t✐♦♥ {c α (Λ, ϑ) , ϑ ∈ Θ} ✇✳r✳t✳ ϑ✳ ❚❤❡♥ ✇❡ ❝❤♦♦s❡ t❤❡ t❤r❡s❤♦•❞ cα = c α Λ, θn ✱ ✇❤❡r❡ θn ✐s s♦♠❡ ❝♦♥s✐st❡♥t ❡st✐♠❛t♦r ♦❢ ϑ ✭s❛②✱ ▼▲❊✮✳ ■♥❞❡❡❞ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t✱ ❢♦r t❤❡ t❡st φn = ✶ ■ { Ṽn>cα} ✱ ✇❡ ❤❛✈❡ lim n→∞ E ϑ φn = α ❢♦r ❛•• ϑ ∈ Θ = (a, b),
✐✳❡✳✱ t❤❡ t❡st φn ✐s ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α✳

❚❤❡r❡ ✐s ❛♥♦t❤❡r ♣♦ss✐❜✐•✐t② t♦ ♦❜t❛✐♥ ❛♥ ❆❉❋ t❡st✳ ■t ✐s ❞♦♥❡ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L[•] ♦❢ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ Ũ (•) ❞❡✜♥❡❞ ❜② ✭✷✳✺✮ s✉❝❤ t❤❛t L[ Ũ ](ν) = w ν ✱ ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❈♦♥s❡q✉❡♥t•②✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ δn = τ 0 L[ Ũn ] (r) 2 Λ( θn , τ ) λ( θn , r) ❞r =⇒ 1 0 w 2 ν ❞ν ≡ ∆, ✇❤❡r❡ Ũn (•) ❣✐✈❡♥ ❜② ✭✷✳✹✮ ✐s t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ Ũ (•)✳ ❋✐♥❛••②✱ ✇❡ ❤❛✈❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢r❡❡ •✐♠✐t ∆✱ t❤❡ t❡st ψn = ✶ ■ { δn > dα} , P{ ∆ > d α } = α ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✳ ❋♦r ❢♦r ♠♦r❡ ❞❡t❛✐•s ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✹✾(✳ ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝♦♥str✉❝t t✇♦ ❆❉❋ •♦❋ t❡sts✱ ❢♦r t❤❡ ❡♠✲ ♣✐r✐❝❛• ❛♥❞ s❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝s✱ ❜❛s❡❞ ♦♥ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✺✾ ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✭✶✳✹✳✶✮✳ ❚❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s ✐s ❣✐✈❡♥ ❜② ✭✷✳✶✮ ❛♥❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✷✱ ✇❡ ♣r♦♣♦s❡ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ t❤❡ ▼❉❊ ϑ * n = arg min ϑ∈Θ Λn (•) -Λ (ϑ, •) , ✇❤❡r❡ Λn (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✷✮ ❛♥❞ • ✐s L 2 (0, τ )✲♥♦r♠✱ ✐✳❡✳✱ Λn (•) -Λ (ϑ, •) 2 = τ 0 Λn (s) -Λ (ϑ, s) 2 ❞s. ■♥ ❙❡❝t✐♦♥ ✷✳✸✱ ✇❡ ❜❡❣✐♥ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ ∆ n = n Λ (ϑ * n , τ ) 2 τ 0 Λn (r) -Λ (ϑ * n , r) 2 λ (ϑ * n , r) ❞r. ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s H 0 ✱ ✇❡ ✜rst s❤♦✇ ✐♥ ▲❡♠♠❛ ✷✳✸✳✹ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r✲ ❣❡♥❝❡ ∆ n =⇒ 1 0 U (ν) 2 ❞ν, ✇❤❡r❡ U (ν) = w (ν) - 1 0 g (ϑ, s) dw (s) ν 0 h (ϑ, s) ds, 1 0 g (ϑ, s) 2 ds = 1. ✭✷✳✻✮ ❚❤❡♥✱ ✐♥ ❙❡❝t✐♦♥ ✷✳✹✱ ✇❡ ✐♥tr♦❞✉❝❡ ❢♦r t❤❡ s❛♠❡ ♠♦❞❡• ♦❢ ♦❜s❡r✈❛t✐♦♥s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❝♦r❡✲❢✉♥❝t✐♦♥ st❛t✐st✐❝ ✭❢♦r♠❛••②✮ ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✸(✮ u * n (t) = 1 √ n n j=1 t 0 λ(ϑ * n , v) λ(ϑ * n , v) [❞X j (v) -λ(ϑ * n , v) ❞v] . ❲❡ s❛② ✏❢♦r♠❛••②✑ ❜❡❝❛✉s❡ t❤❡ ▼❉❊ ϑ * n ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❤♦•❡ tr❛•❡❝t♦r② X n ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• t 0 λ(ϑ * n , v) λ(ϑ * n , v) ❞X j (v) ✐s ♥♦t ✇❡•• ❞❡✜♥❡❞✳ ❚❤❡ ❝♦rr❡❝t ❞❡✜♥✐t✐♦♥ ✇✐•• ❜❡ ❣✐✈❡♥ •❛t❡r ✉s✐♥❣ ❛♥ ❛♣♣r♦❛❝❤ ✇❤✐❝❤ ✇❛s ❛♣♣•✐❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✐♥ ❑✉t♦②❛♥ts ❬✹✸(✳ ❚❤❡ s❝♦r❡✲❢✉♥❝t✐♦♥ st❛t✐st✐❝ u * n (•) ❝♦♥✈❡r❣❡s t♦ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ u (•) ❞❡✜♥❡❞ ❜② ✭✷✳✸✽✮ ❜❡•♦✇✳ ❍❡♥❝❡✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✷✳✸✾✮ •❡❛❞s t♦ t❤❡ •❛✉ss✐❛♥ ♣r♦❝❡ss U (•) ❣✐✈❡♥ ❜② ✭✷✳✹✷✮ ✭s❡❡ ▲❡♠♠❛ ✷✳✹✳✺ ❜❡•♦✇✮✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✐♥ ❜♦t❤ ❝❛s❡s✱ ✐✳❡✳✱ t❤❡ ❡♠♣✐r✐❝❛• ❛♥❞ s❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝s✱ ✇❡ ♦❜t❛✐♥ t❤❡ s❛♠❡ •✐♠✐t ♣r♦❝❡ss U (•) ✇✐t❤ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s ✻✵ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ h (•, •) ❛♥❞ g (•, •)✳ ◆♦✇ t♦ ♦❜t❛✐♥ t❤❡ ❆❉❋ •✐♠✐t ♣r♦❝❡ss❡s ✇❡ ❛♣♣•② t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L[•] ❞❡✜♥❡❞ ❜② ✭✶✳✷✻✮ t♦ t❤❡ ♣r♦❝❡ss U (•) s✉❝❤ t❤❛t L[U ] (ν) = w ν , 0 ≤ ν ≤ 1. ❚❤❡ ❛❜♦✈❡ ♣r❡s❡♥t❛t✐♦♥ s✉❣❣❡sts t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❡sts ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ U n (•, •) ❛♥❞ U * n (•) ♦❢ U (•) ✭❣✐✈❡♥ ❜② ✭✷✳✶✷✮ ❛♥❞ ✭✷✳✹✸✮✱ r❡s♣❡❝✲ t✐✈❡•② ❜❡•♦✇✮✳ ❚❤❡♥✱ ✇❡ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L [•] t♦ ❜♦t❤ st❛t✐st✐❝s U n (•, •) ❛♥❞ U * n (•) ❛♥❞ ✇❡ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss ✭✉♥❞❡r H 0 ✮ L[U n ](r) =⇒ L[U ] (ν) = w ν , L[U * n ](ν) =⇒ L[U ] (ν) = w ν , 0 ≤ ν ≤ 1. ■♥tr♦❞✉❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛t✐st✐❝s ❢♦r t❤❡ ❡♠♣✐r✐❝❛• ❛♥❞ s❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝s✱ r❡s♣❡❝t✐✈❡•② ∆ * n = τ 0 L[U n ](r) 2 Λ (ϑ * n , τ ) λ (ϑ * n , r) ❞r, δn = 1 0 L[U * n ](ν) 2 ❞ν. ■♥❞❡❡❞✱ ✇❡ ✇✐•• s❤♦✇ t❤❛t ❢♦r ❜♦t❤ st❛t✐st✐❝s ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✉♥❞❡r H 0 ✮ ∆ * n =⇒ 1 0 w 2 ν ❞ν, δn =⇒ 1 0 w 2 ν ❞ν. ❚❤❡r❡❢♦r❡ t❤❡ t❡sts ψ * n = ✶ ■ { ∆ * n >cα} , φn = ✶ ■ { δn>cα} , ✇✐t❤ t❤❡ t❤r❡s❤♦•❞ c α s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ P 1 0 w 2 ν ❞ν > c α = α, ✇✐•• ❜❡ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✱ ✐✳❡✳✱ t❤❡② ❜❡•♦♥❣ t♦ t❤❡ ❝•❛ss K α = φ n : lim n→∞ E ϑ φ n = α, ∀ϑ ∈ Θ = (a, b) ❛♥❞ ✇✐•• ❜❡ ❆❉❋✳ ❲❡ ♣r♦♣♦s❡ t❤❡s❡ t❡sts ✐♥ ❚❤❡♦r❡♠ ✷✳✸✳✸ ❛♥❞ ✷✳✹✳✹ ❜❡•♦✇✳ ✷✳✷ ▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r ❙✉♣♣♦s❡ t❤❛t ✇❡ ❤❛✈❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❛ ♣❡r✐♦❞✐❝ P♦✐ss♦♥ ♣r♦❝❡ss X n = (X t , 0 ≤ t ≤ T = nτ ) ✭✷✳✼✮ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✻✶ ♦❢ ❦♥♦✇♥ ♣❡r✐♦❞ τ > 0. ❲❡ ❤❛✈❡ ❢♦r 0 ≤ s < t ❛♥❞ k = 0, 1, 2, . . . P{X t -X s = k} = Λ (s, t) k k! e -Λ(s,t) , Λ (s, t) = t s λ (v) ❞v. ❲❡ s❡❡ t❤❛t X t -X s ✐s ❛ P♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜•❡ ✇✐t❤ ♠❡❛♥ Λ (s, t)✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡ ♠❡❛♥ ❛♥❞ ✐♥t❡♥s✐t② ❢✉♥❝t✐♦♥s ♦❢ X t ❛r❡ ❣✐✈❡♥ r❡s♣❡❝t✐✈❡•② ❜② Λ(t) = Λ (0, t) ❛♥❞ λ (t + kτ ) = λ (t)✳ ❖✉r ❣♦❛• ✐s t♦ ❝♦♥str✉❝t •♦❋ t❡sts ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 : Λ(•) ∈ {Λ(ϑ, •), ϑ ∈ Θ = (a, b)} , ✇❤❡r❡ Λ (ϑ, t) = t 0 λ (ϑ, v) ❞v ✐s ❛ ❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ▼❉❊ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ ϑ * n = arg min ϑ∈Θ Λn (•) -Λ (ϑ, •) , ✇❤❡r❡ Λn (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✷✮ ❛♥❞ • ✐s L 2 (0, τ )✲♥♦r♠✱ ✐✳❡✳✱ Λn (•) -Λ (ϑ, •) 2 = τ 0 Λn (s) -Λ (ϑ, s) 2 ❞s. ■♥tr♦❞✉❝❡ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✳ R 0 ✳ ❚❤❡ ✐♥t❡♥s✐t② ❢✉♥❝t✐♦♥ λ(t), 0 ≤ t ≤ T ♦❢ t❤❡ ♦❜s❡r✈❡❞ ♣r♦❝❡ss X n ❜❡•♦♥❣s t♦ t❤❡ ♣❛r❛♠❡tr✐❝ ❢❛♠✐•② {(λ(ϑ, t), 0 ≤ t ≤ T ) , ϑ ∈ Θ = (a, b)}, ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ϑ 0 ∈ Θ s✉❝❤ t❤❛t λ(t) = λ(ϑ 0 , t) ❢♦r ❛•• t ∈ [0, T ]. R 1 ✳ ❚❤❡ ✐♥t❡♥s✐t② ❢✉♥❝t✐♦♥ λ(ϑ, t) ✐s str✐❝t•② ♣♦s✐t✐✈❡✱ ❤❛s t✇♦ ❝♦♥t✐♥✉♦✉s ❜♦✉♥❞❡❞ ❞❡r✐✈❛t✐✈❡s ✇✳r✳t✳ ϑ ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡s λ(ϑ, t) ❛♥❞ λ(ϑ, t) ❛r❡ ❛•s♦ τ ✲ ♣❡r✐♦❞✐❝✳ R 2 ✳ ❚❤❡ ❋✐s❤❡r ✐♥❢♦r♠❛t✐♦♥ ✭✷✳✸✮ ✐s str✐❝t•② ♣♦s✐t✐✈❡✳ R 3 ✳ ❚❤❡ ✐❞❡♥t✐✜❛❜✐•✐t② ❝♦♥❞✐t✐♦♥ ✐s ❛s ❢♦••♦✇s ✿ ❢♦r ❛♥② ν > 0 ❛♥❞ ϑ 0 ∈ Θ g (ν) ≡ inf ϑ:|ϑ-ϑ 0 |>ν τ 0 [Λ(ϑ, t) -Λ(ϑ 0 , t)] 2 ❞t > 0. ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ❝♦♥❞✐t✐♦♥s ♦❢ t❤✐s t②♣❡ ❛♣♣❡❛r ✐♥ ❛•♠♦st ❛•• st❛t✐st✐❝❛• ♣r♦❜•❡♠s ♦❢ ♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ❢♦r r❡❣✉•❛r ❝❛s❡ ✭s❡❡✱ ❡✳❣✳✱ ✐♥ ❬✹✺(✮✳ ■♥ t❤❡ ✻✷ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ♣r❡s❡♥t❛t✐♦♥ ❜❡•♦✇✱ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ ♥✉•• ❤②♣♦t❤❡s✐s H 0 ❛r❡ ❛•✇❛②s ❢✉•✜••❡❞✳ ❚❤❡ ♣r♦♣❡rt✐❡s ❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ (n → ∞) ❜❡❤❛✈✐♦r ♦❢ t❤❡ ▼❉❊ ❢♦r t❤✐s ♠♦❞❡• ✇❡r❡ st✉❞✐❡❞ ✉♥❞❡r r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ✐♥ ❬✹✺(✳ ❚❤❡ ▼❉❊ ϑ * n s❛t✐s✜❡s t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡q✉❛t✐♦♥ ✭▼❉❊q✮ ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✺(✮ τ 0 Λn (s) -Λ (ϑ * n , s) Λ (ϑ * n , s) ❞s = 0. ✭✷✳✽✮ ❉❡♥♦t❡ u * * n = √ n (ϑ * n -ϑ)✳ ❇② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ Λ(ϑ * n , s)-Λ(ϑ, s) = u * * n √ n Λ( θ, s)✱ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✽✮ ❜❡❝♦♠❡s τ 0 Λn (s) -Λ (ϑ, s) - u * * n √ n Λ( θ, s) Λ (ϑ * n , s) ❞s = 0, ✇❤❡r❡ | θ -ϑ| ≤ |ϑ * n -ϑ|✳ ■♥❞❡❡❞ ❛❢t❡r r❡♣•❛❝✐♥❣ Λn (•) ❜② ✭✷✳✷✮✱ ✇❡ ♦❜t❛✐♥ u * * n = τ 0 Λ (ϑ * n , s) Λ( θ, s) ❞s -1 1 √ n n j=1 τ 0 (X j (s) -Λ (ϑ, s)) Λ (ϑ * n , s) ❞s = τ 0 Λ(ϑ * n , s) Λ( θ, s) ❞s -1 1 √ n n j=1 τ 0 s 0 ❞π j (v) Λ(ϑ * n , s) ❞s. ❍❡r❡ ✇❡ ❞❡✜♥❡ t❤❡ ❝❡♥t❡r❡❞ P♦✐ss♦♥ ♣r♦❝❡ss ❜② π j (v) = X j (v) -Λ (ϑ, v)✳ ❯s✐♥❣ ❋✉❜✐♥✐✬s t❤❡♦r❡♠✱ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜•❡ u * * n ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✿ u * * n = τ 0 Λ (ϑ * n , s) Λ( θ, s) ❞s -1 1 √ n n j=1 τ 0 τ v Λ (ϑ * n , s) ❞s ❞π j (v). ❇② t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ✭s❡❡✱ ❡✳❣✳✱ ❬✸✽(✮✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ 1 √ n n j=1 τ 0 τ v Λ (ϑ * n , s) ❞s ❞π j (v) =⇒ τ 0 τ v Λ (ϑ, s) ❞s λ (ϑ, v) ❞W v , ✇❤❡r❡ W v , 0 ≤ v ≤ τ ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡r❡❢♦r❡ ✇❡ ♦❜t❛✐♥ √ n (ϑ * n -ϑ) =⇒ J (ϑ) -1 τ 0 τ v Λ (ϑ, s) ❞s λ (ϑ, v) ❞W v , ✭✷✳✾✮ ✇❤❡r❡ J (ϑ) = τ 0 Λ (ϑ, s) 2 ❞s. ✭✷✳✶✵✮ ✻✸ ❍❡♥❝❡✱ ✉♥❞❡r t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✱ ✇❡ ❤❛✈❡ t❤❡ ❛s②♠♣t♦t✐❝ ♥♦r♠❛•✐t② L ϑ { √ n (ϑ * n -ϑ)} =⇒ L{ξ} = N 0, σ (ϑ) 2 . ❍❡r❡ t❤❡ ✈❛r✐❛♥❝❡ •✐♠✐t ✐s ❛s ❢♦••♦✇s ✿ σ (ϑ) 2 = J (ϑ) -2 τ 0 τ v Λ (ϑ, s) ❞s 2 λ (ϑ, v) ❞v. ✭✷✳✶✶✮ ❖✉r ❛✐♠ ✐s t♦ ❝♦♥str✉❝t •♦❋ t❡sts✱ ✇❤✐❝❤ ❛r❡ ❆❉❋✱ ✐✳❡✳✱ ✇❡ s❡❡❦ t❤❡ t❡sts st❛t✐st✐❝s ✇❤♦s❡ •✐♠✐ts ❞✐str✐❜✉t✐♦♥ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❞❡r•②✐♥❣ ♠♦❞❡• ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥ Λ (•, •) ❛♥❞ ♣❛r❛♠❡t❡r ϑ. ■♥❞❡❡❞✱ ✇❡ ❛♣♣•② t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✷✻✮ t♦ t❤❡ st❛t✐st✐❝ U (•) ✭✷✳✻✮ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❆❉❋ •♦❋ t❡sts✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✇❡ ✇✐•• tr❡❛t t❤❡ ♣r♦❜•❡♠ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t✇♦ t❡sts✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛• ❜❛s✐❝ st❛t✐st✐❝ ✭❙❡❝t✐♦♥ ✷✳✸✮ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ♦♥ t❤❡ s❝♦r❡✲❢✉♥❝t✐♦♥ st❛t✐st✐❝ ✭❙❡❝t✐♦♥ ✷✳✹✮✳ ✷✳✸ ❊♠♣✐r✐❝❛• ❜❛s✐❝ st❛t✐st✐❝ ❲❡ ❜❡❣✐♥ ❜② ❝♦♥str✉❝t✐♥❣ t❤❡ ❆❉❋ •♦❋ t❡st ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛• ❜❛s✐❝ st❛t✐st✐❝✳ ❈♦♥s✐❞❡r t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ ∆ n = τ 0 U n (ϑ * n , r) 2 Λ (ϑ * n , τ ) λ (ϑ * n , r) ❞r. ❍❡r❡ U n (ϑ * n , r) = ζ n (ϑ * n , r) Λ (ϑ * n , τ ) , ζ n (ϑ * n , r) = √ n Λn (r) -Λ (ϑ * n , r) , ✭✷✳✶✷✮ ✇❤❡r❡ ϑ * n ✐s t❤❡ ▼❉❊ ❛♥❞ Λn (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✷✮✳ ❇❡•♦✇ ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ •✐♠✐t st❛t✐st✐❝ ♦❢ ∆ n ❞❡♣❡♥❞s str♦♥❣•② ♦♥ t❤❡ ♠♦❞❡• ❛♥❞ t❤❡ ♣❛r❛♠❡t❡r ϑ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✷✻✮ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❆❉❋ •✐♠✐t st❛t✐st✐❝✳ ❉❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥s g (ϑ, s) = C 1 (ϑ) -1 2 τ v(s) Λ (ϑ, z) ❞z ✭✷✳✶✸✮ ✻✹ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❛♥❞ h (ϑ, s) = C 1 (ϑ) 1 2 J (ϑ) -1 Λ (ϑ, τ ) λ(ϑ, v(s)) λ(ϑ, v(s)) , ✭✷✳✶✹✮ ✇❤❡r❡ v(s) ✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ s = Λ (ϑ, v) Λ (ϑ, τ ) -1 ✱ J (ϑ) ✐s ❣✐✈❡♥ ❜② ✭✷✳✶✵✮ ❛♥❞ 1 0 g (ϑ, s) 2 ❞s = 1, C 1 (ϑ) = 1 0 τ v(s) Λ (ϑ, z) ❞z 2 ❞s.
▲❡t ✉s ❞❡♥♦t❡ t❤❡ •❛✉ss✐❛♥ ♣r♦❝❡ss

U (ν) = w(ν) - 1 0 g (ϑ, s) ❞w(s) ν 0 h (ϑ, s) ❞s, ✭✷✳✶✺✮ ✇❤❡r❡ w(ν), 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t✳ ▲❡♠♠❛ ✷✳✸✳✹✳ ▲❡t t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ❜❡ ❢✉•✜••❡❞✱ t❤❡♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∆ n =⇒ ∆ = 1 0 U (ν) 2 ❞ν ✭✷✳✶✻✮ ❤♦•❞s✱ ✉♥❞❡r H 0 ✳ Pr♦♦❢✳ ▲❡t ✉s ✜rst s❤♦✇ t❤❛t ✭✉♥❞❡r H 0 ✮ U n (ϑ * n , r) =⇒ U (ν), 0 ≤ ν ≤ 1. ✭✷✳✶✼✮ ❲❡ ♣r♦✈❡ t❤✐s ❛s ❢♦••♦✇s✳ ❇② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ t❤❡ ♣r♦❝❡ss ζ n (•, •) ❞❡✜♥❡❞ ❜② ✭✷✳✶✷✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ζ n (ϑ * n , r) = √ n Λn (r) -Λ (ϑ * n , r) = √ n Λn (r) -Λ (ϑ, r) - √ n (ϑ * n -ϑ) Λ( θ, r) + o(1). ❚❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ②✐❡•❞s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ √ n Λn (r) -Λ (ϑ, r) =⇒ W (Λ (ϑ, r)) ∼ N (0, Λ (ϑ, r)) , ✇❤❡r❡ W (•) ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❯♥❞❡r t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ √ n (ϑ * n -ϑ) =⇒ J(ϑ) -1 τ 0 τ v Λ(ϑ, z) ❞z ❞W (Λ(ϑ, v)) ∼ N (0, σ(ϑ) 2 ), ✭✷✳✶✽✮ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✻✺ ✇❤❡r❡ J (ϑ) ❛♥❞ σ (ϑ) 2 ❛r❡ ❣✐✈❡♥ ❜② ✭✷✳✶✵✮-✭✷✳✶✶✮✱ r❡s♣❡❝t✐✈❡•②✳ ❈♦♥s❡q✉❡♥t•②✱ ✉♥❞❡r H 0 ✱ ✇❡ ❣❡t U n (ϑ * n , r) = ζ n (ϑ * n , r) Λ (ϑ * n , τ ) =⇒ W (Λ (ϑ, r)) Λ (ϑ, τ ) -J (ϑ) -1 τ 0 τ v Λ (ϑ, z) ❞z ❞W (Λ (ϑ, v)) Λ(ϑ, r) Λ (ϑ, τ ) ≡ u (r) . ✭✷✳✶✾✮ ❋✉rt❤❡r✱ ✇❡ ❝❛♥ ✇r✐t❡ Λ(ϑ, r) = r 0 λ(ϑ, v) ❞v = r 0 λ(ϑ, v) λ(ϑ, v) ❞Λ (ϑ, v) . ❍❡♥❝❡ t❤❡ •✐♠✐t ♣r♦❝❡ss u (•) ❞❡✜♥❡❞ ❜② ✭✷✳✶✾✮ ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ u (r) ≡ W (Λ (ϑ, r)) Λ (ϑ, τ ) -J (ϑ) -1 Λ (ϑ, τ ) τ 0 τ v Λ (ϑ, z) ❞z ❞W (Λ (ϑ, v)) Λ (ϑ, τ ) r 0 λ(ϑ, v) λ(ϑ, v) ❞Λ (ϑ, v) Λ (ϑ, τ ) . ✭✷✳✷✵✮ ▲❡t ✉s ❝❤❛♥❣❡ t❤❡ ✈❛r✐❛❜•❡s s = Λ(ϑ, v) Λ(ϑ, τ ) , ν = Λ(ϑ, r) Λ(ϑ, τ ) , w(ν) = W (Λ(ϑ, r)) Λ(ϑ, τ ) , w(s) = W (Λ(ϑ, v)) Λ(ϑ, τ ) . ✭✷✳✷✶✮ ❚❤❡♥✱ U n (ϑ * n , r) =⇒ w(ν) -J (ϑ) -1 Λ (ϑ, τ ) 1 0 τ v(s) Λ (ϑ, z) ❞z ❞w(s) ν 0 λ(ϑ, v(s)) λ(ϑ, v(s)) ❞s = w(ν) - 1 0 g (ϑ, s) ❞w(s) ν 0 h (ϑ, s) ❞s = U (ν), ✇❤❡r❡ v(s) ✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ s = Λ (ϑ, v) Λ (ϑ, τ ) -1 ✱ g (•, •) ❛♥❞ h (•, •) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✷✳✶✸✮ ❛♥❞ ✭✷✳✶✹✮✱ r❡s♣❡❝t✐✈❡•②✳ ❲❡ ✜♥❛••② ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✷✳✶✼✮ ✉♥❞❡r H 0 ✳ ▲❡t Z n (•, •) ❛♥❞ Z (•) ❜❡ ❣✐✈❡♥ ❜② Z n (ϑ * n , r) = U n (ϑ * n , r) 2 , Z (ν) = U (ν) 2 . ◆♦✇ ✐♥ ♦r❞❡r t♦ ❣❡t t❤❡ ✜♥❛• r❡s✉•t ✭✷✳✶✻✮✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♥t❡❣r❛•s✱ ✐✳❡✳✱ ✇❡ ❤❛✈❡ t♦ ✈❡r✐❢② t❤❡ ❢♦••♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿ ✭s❡❡ ❬✷✼(✱ ❚❤❡♦r❡♠ ❆✶✳✷✷✮ ✻✻ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✶✳ ❋♦r ❛♥② r 1 , . . . , r k ❛♥❞ ν 1 , . . . , ν k ✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✈❡❝t♦rs (Z n (ϑ * n , r 1 ) , . . . , Z n (ϑ * n , r k )) =⇒ (Z (ν 1 ) , . . . , Z (ν k )) ✭✷✳✷✷✮ ❤♦•❞s✳ ✷✳ ❋♦r r i ∈ [0, τ ], i = 1, 2✱ E ϑ |Z n (ϑ * n , r 1 ) -Z n (ϑ * n , r 2 )| ≤ C |r 1 -r 2 | 1 2 , ✭✷✳✷✸✮ ✇❤❡r❡ C > 0 ✐s s♦♠❡ ❝♦♥st❛♥t✳ ✸✳ ❋♦r ❛♥② κ > 0✱ sup n,ϑ E ϑ Z n (ϑ * n , r) < κ. ✭✷✳✷✹✮ ❇② ✉s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts t♦ ♣r♦✈❡ ✭✷✳✶✼✮✱ ❢♦r ❛♥② r❡❛• ♥✉♠❜❡rs v 1 , . . . , v k ✱ r 1 , . . . , r k ❛♥❞ ν 1 , . . . , ν k ✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✈❡❝t♦rs k l=1 v l U n (ϑ * n , r l ) =⇒ k l=1 v l U (ν l ) . ❚❤❡r❡❢♦r❡✱ ❜② ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ t❤❡♦r❡♠✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛• ❞✐str✐❜✉t✐♦♥s (Z n (ϑ * n , r 1 ) , . . . , Z n (ϑ * n , r k )) =⇒ (Z (ν 1 ) , . . . , Z (ν k )) . ▼♦r❡♦✈❡r✱ ✉s✐♥❣ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t②✱ ✇❡ ❝❛♥ ✇r✐t❡ E ϑ |Z n (ϑ * n , r 1 ) -Z n (ϑ * n , r 2 )| 2 = E ϑ |U n (ϑ * n , r 1 ) -U n (ϑ * n , r 2 )| |U n (ϑ * n , r 1 ) + U n (ϑ * n , r 2 )| 2 ≤ E ϑ |U n (ϑ * n , r 1 ) -U n (ϑ * n , r 2 )| 2 E ϑ |U n (ϑ * n , r 1 ) + U n (ϑ * n , r 2 )| 2 . ❋♦r t❤❡ ✜rst ♠❛t❤❡♠❛t✐❝❛• ❡①♣❡❝t❛t✐♦♥✱ ❜② ❛♣♣•②✐♥❣ ❚❛②•♦r✬s ❢♦r♠✉•❛✱ ✇❡ ❣❡t E ϑ √ n Λn (r 1 ) -Λ (ϑ * n , r 1 ) - √ n Λn (r 2 ) -Λ (ϑ * n , r 2 ) 2 = 1 n E ϑ n j=1 r 1 r 2 [❞X j (r) -λ (ϑ * n , r) ❞r] 2 ≤ 2 n E ϑ n j=1 r 1 r 2 [❞X j (r) -λ(ϑ, r)❞r] 2 + 2 n E ϑ n j=1 r 1 r 2 [λ(ϑ, r) -λ(ϑ * n , r)]❞r ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✻✼ ≤ 2 n n j=1 E ϑ r 1 r 2 ❞π j (r) 2 + 2 n E ϑ n j=1 r 1 r 2 (ϑ -ϑ * n ) λ( θ, r) ❞r 2 ≤ 2 r 1 r 2 λ (ϑ, r) ❞r + 2 n E ϑ (ϑ * n -ϑ) 2 r 1 r 2 λ( θ, r) ❞r 2 . ❇② t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ✱ E ϑ |U n (ϑ * n , r 1 ) -U n (ϑ * n , r 2 )| 2 ≤ C |r 1 -r 2 | . ❙✐♠✐•❛r•②✱ ❢♦r t❤❡ s❡❝♦♥❞ ♠❛t❤❡♠❛t✐❝❛• ❡①♣❡❝t❛t✐♦♥✱ ✇❡ ❤❛✈❡ ❢♦r r i ∈ [0, τ ], i = 1, 2 E ϑ √ n Λn (r 1 ) -Λ (ϑ * n , r 1 ) + √ n Λn (r 2 ) -Λ (ϑ * n , r 2 ) 2 ≤ 2 n E ϑ n j=1 r 1 0 [❞X j (r) -λ(ϑ * n , r)❞r] 2 + 2 n E ϑ n j=1 r 2 0 [❞X j (r) -λ(ϑ * n , r)❞r] 2 ≤ 4 n E ϑ n j=1 r 1 0 [❞X j (r) -λ(ϑ, r)❞r] 2 + 4 n E ϑ n j=1 r 2 0 [❞X j (r) -λ(ϑ, r)❞r] 2 + 4 n E ϑ n j=1 r 1 0 (ϑ -ϑ * n ) λ( θ, r) ❞r 2 + 4 n E ϑ n j=1 r 2 0 (ϑ -ϑ * n ) λ( θ, r) ❞r 2 ≤ 4 r 1 0 λ (ϑ, r) ❞r + 4 r 2 0 λ (ϑ, r) ❞r + 4 n E ϑ (ϑ * n -ϑ) 2 r 1 0 λ( θ, r) ❞r 2 + 4 n E ϑ (ϑ * n -ϑ) 2 r 2 0 λ( θ, r) ❞r 2 . ❍❡♥❝❡✱ ❜② r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ✱ E ϑ |U n (ϑ * n , r 1 ) + U n (ϑ * n , r 2 )| 2 ≤ C. ❋r♦♠ ✇❤❛t ❤❛s ❛•r❡❛❞② ❜❡❡♥ ♣r♦✈❡❞✱ ✐t ❢♦••♦✇s t❤❛t t❤❡ ❡st✐♠❛t❡ ✭✷✳✷✸✮ ❤♦•❞s✳ ◆♦✇ t♦ ❝❤❡❝❦ ✭✷✳✷✹✮✱ ✇❡ ❝❛♥ ✇r✐t❡ E ϑ √ n Λn (r) -Λ (ϑ * n , r) 2 ≤ 2 E ϑ √ n Λn (r) -Λ (ϑ, r) 2 + 2 E ϑ √ n (Λ (ϑ, r) -Λ (ϑ * n , r)) 2 ≤ 2 E ϑ (X 1 (r) -Λ (ϑ, r)) 2 + 2 n E ϑ (ϑ * n -ϑ) 2 Λ( θ, r) 2 ≤ 2 Λ (ϑ, r) + C Λ( θ, r) 2 . ❲❡ t❤✉s ❣❡t ✭✷✳✷✹✮ ❜② t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ✳ ✻✽ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❋✐♥❛••②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ❞✐str✐❜✉t✐♦♥s ✭✷✳✷✷✮✱ t❤❡ ❡st✐✲ ♠❛t❡ ✭✷✳✷✸✮ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✷✳✷✹✮ ❛••♦✇ ✉s t♦ ✈❡r✐❢② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✷✳✶✻✮ ❛♥❞ t❤❡ ▲❡♠♠❛ ✐s ♣r♦✈❡❞✳ ■t ✐s ♦❜✈✐♦✉s t❤❛t t❤❡ •✐♠✐t ∆ = 1 0 U (ν) 2 ❞ν, ✇❤❡r❡ U (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✶✺✮✱ ❞❡♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡• Λ (•, •) ❛♥❞ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❇② ❛♣♣•②✐♥❣ t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L[•] ✭✶✳✷✻✮ t♦ t❤❡ st✉❞✐❡❞ ♣r♦❝❡ss U (•)✱ ✇❡ ♦❜t❛✐♥ 1 0 L[U ](ν) 2 ❞ν = 1 0 w 2 ν ❞ν. ✭✷✳✷✺✮ ■♥ ✈✐❡✇ ♦❢ t❤❡ ❛❜♦✈❡ ♣r♦♣❡rt②✱ ✇❡ ✇✐•• ✐♥tr♦❞✉❝❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✇✐t❤ t❤❡ s❛♠❡ •✐♠✐t ✭✷✳✷✺✮✳ ■♥❞❡❡❞ ✇❡ ✇✐•• s❤♦✇ t❤❛t ∆ * n = τ 0 L[U n ](r) 2 Λ (ϑ * n , τ ) λ (ϑ * n , r) ❞r =⇒ 1 0 w 2 ν ❞ν ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ t❡st ψ * n = ✶ ■ { ∆ * n >cα} ✇✐•• ❜❡ ❆❉❋✳ ❉❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥s g (ϑ, v) = C1 (ϑ) -1 2 τ v Λ (ϑ, z) ❞z ✭✷✳✷✻✮ ❛♥❞ h (ϑ, v) = C1 (ϑ) 1 2 J (ϑ) -1 Λ (ϑ, τ ) λ(ϑ, v) λ(ϑ, v) . ✭✷✳✷✼✮ ❍❡r❡ J (ϑ) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✶✵✮ ❛♥❞ C1 (ϑ) = τ 0 τ q Λ (ϑ, z) ❞z 2 ❞Λ(ϑ, q) Λ(ϑ, τ ) . ❚❤❡♥ ❞❡♥♦t❡ Ĩ1 = 1 Λ(ϑ, τ ) s 0 g (ϑ, v) 2 ❞Λ(ϑ, v), Ĩ2 = 1 Λ(ϑ, τ ) s 0 h (ϑ, v) g (ϑ, v) ❞Λ(ϑ, v), Ĩ3 = 1 Λ(ϑ, τ ) s 0 h (ϑ, v) ❞Λ(ϑ, v), Ĩ4 = 1 Λ(ϑ, τ ) s 0 h (ϑ, v) 2 ❞Λ(ϑ, v) ❛♥❞ ♣✉t Ĩ5 = 1 Λ(ϑ, τ ) s 0 g (ϑ, v) ❞Λ(ϑ, v). ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✻✾ ❚❤✐s ❛••♦✇s ✉s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥s φ1 (•, •), φ2 (•, •) ❛♥❞ ψ2 (•, •) ❣✐✈❡♥ ❜② ✭✶✳✷✸✮✱ ✭✶✳✷✹✮ ❛♥❞ ✭✶✳✷✺✮✱ r❡s♣❡❝t✐✈❡•②✱ ✇❤❡r❡ ✐♥st❡❛❞ ♦❢ g ❛♥❞ h✱ ✇❡ ✉s❡ g = g (ϑ, s) ❛♥❞ h = h (ϑ, s)✳ ■♥ t❤❡ s❡q✉❡• ✇❡ r❡♣•❛❝❡ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ ❜② t❤❡ ▼❉❊ ϑ * n ✐♥ g (•, •)✱ h (•, •)✱ φ1 (•, •), φ2 (•, •) ❛♥❞ ψ2 (•, •) ❛♥❞ ✇❡ ❞❡✜♥❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ gn (•, •)✱ hn (•, •)✱ φ1,n (•, •), φ2,n (•, •) ❛♥❞ ψ2,n (•, •)✱ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡st✐♠❛t♦r ϑ * n ❛♥❞ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ❣✐✈❡ ✉s t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ (n → ∞) sup s∈[0,τ ] | φ2,n (ϑ * n , s) -φ2 (ϑ, s)| -→ 0. ❍❡♥❝❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ φ+ 2,n (ϑ * n , s) =    φ2,n (ϑ * n , s) -1 , ✐❢ φ2,n (ϑ * n , s) > 0, 0, ❡•s❡. ✭✷✳✷✽✮ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❝♦✐♥❝✐❞❡s ❛s②♠♣t♦t✐❝❛••② ✇✐t❤ φ2 (ϑ, s) -1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❝❤❛♥❣❡❞✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ ✭❢♦r♠❛••②✮ t❤❡ ❢♦••♦✇✐♥❣ st❛t✐st✐❝ Ŵn (r) = U n (ϑ * n , r) + 1 Λ (ϑ * n , τ ) r 0 s 0 φ+ 2,n (ϑ * n , s) φ1,n (ϑ * n , s) hn (ϑ * n , q) + ψ2,n (ϑ * n , s) gn (ϑ * n , q) ❞U n (ϑ * n , q) λ (ϑ * n , s) ❞s. ✭✷✳✷✾✮ ❍❡r❡ t❤❡ ♣r♦❝❡ss U n (•, •) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✶✷✮✳ ■♥ ♦r❞❡r t♦ ❣❡t t❤❡ ❆❉❋ •♦❋ t❡st✱ ✐t ✇✐•• ❜❡ ♥❡❝❡ss❛r② t♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ Ŵn (r) =⇒ L [U ] (ν) = w ν . ❚❤✉s t❤❡ t❡st ❜❛s❡❞ ♦♥ Ŵn (•) ✇✐•• ❜❡ ❆❉❋✳ ■t ✐s ❡❛s✐•② s❡❡♥ t❤❛t t❤❡ ♣r♦❝❡ss A n (ϑ * n , s) = s 0 gn (ϑ * n , q) ❞U n (ϑ * n , q) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ A n (ϑ * n , s) = 1 √ n Λ (ϑ * n , τ ) n j=1 s 0 gn (ϑ * n , q) ❞X j (q) - √ n Λ (ϑ * n , τ ) s 0 gn (ϑ * n , q) λ (ϑ * n , q) ❞q. ✭✷✳✸✵✮ ✼✵ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ■t ✐s ❝•❡❛r t❤❛t ✇❡ ❝❛♥ ♥♦t ❝❛•❝✉•❛t❡ t❤✐s ♣r♦❝❡ss ❞✐r❡❝t•②✱ ❜❡❝❛✉s❡ ✐♥ t❤❡ ✜rst ✐♥t❡✲ ❣r❛• t❤❡ ✐♥t❡❣r❛♥❞ ❝♦♥t❛✐♥s t❤❡ ▼❉❊ ϑ * n ✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❤♦•❡ tr❛•❡❝t♦r② X n = (X t , 0 ≤ t ≤ T = nτ )✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• s 0 gn (ϑ * n , q) ❞X j (q) ✭✷✳✸✶✮ ✐s ♥♦t ✇❡•• ❞❡✜♥❡❞✳ ■♥❞❡❡❞✱ t♦ ❛✈♦✐❞ t❤✐s ♣r♦❜•❡♠✱ ✇❡ ✉s❡ ❛♥ ❛♣♣r♦❛❝❤✱ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ s♦♠❡ ♣r❡•✐♠✐♥❛r② ❝♦♥s✐st❡♥t ❡st✐♠❛t♦r θN ✳ ◆♦t❡ t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ✇❛s ❛♣♣•✐❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✐♥ ❬✹✸(✳ ❍❡♥❝❡ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r t✇♦ ❞✐✛❡r❡♥t ❡st✐♠❛t♦rs ✐♥ t❤❡ ❝❛•❝✉•❛t✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss A n (•, •) ❛s ❢♦••♦✇s ✿ ❋♦r t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✭✷✳✸✶✮✱ ✐t ✐s s✉✣❝✐❡♥t ❢♦r θN t♦ ❜❡ ❝♦♥s✐st❡♥t ❛♥❞ ❢♦r t❤❡ s❡❝♦♥❞ ♦r❞✐♥❛r② ✐♥t❡❣r❛• s 0 gn (ϑ * n , q) λ (ϑ * n , q) ❞q ✇❡ ♥❡❡❞ t❤❡ ❛s②♠♣t♦t✐❝ ♥♦r♠❛•✐t② ♦❢ ϑ * n ✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r ❝♦♥str✉❝t❡❞ ❜② t❤❡ ✜rst N = [ √ n] ♦❜s❡r✈❛t✐♦♥s X N = (X 1 , . . . , X N ) θN = arg min ϑ∈Θ τ 0 ΛN (t) -Λ (ϑ, t) 2 ❞t. ✭✷✳✸✷✮ ❍❡r❡ N ✐s t❤❡ ❡♥t✐r❡ ♣❛rt ♦❢ √ n ❛♥❞ ΛN (•) ✐s t❤❡ ❡♠♣✐r✐❝❛• ♠❡❛♥ ❢✉♥❝t✐♦♥ ΛN (t) = 1 N N j=1 X j (t), ✇❤❡r❡ X j (t) = X τ (j-1)+t -X τ (j-1) , t ∈ [0, τ ]. ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t t❤✐s ❡st✐♠❛t♦r ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• ✉♥❞❡r r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ✭s❡❡ ❬✹✺( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮✳ ❈♦♥s❡q✉❡♥t•②✱ ✇❡ ♠♦❞✐❢② t❤❡ st❛t✐st✐❝ A n (•, •) ❣✐✈❡♥ ❜② ✭✷✳✸✵✮ ❛s ❢♦••♦✇s ✿ Ân θN , s = 1 √ n Λ θN , τ n j=N +1 s 0 gn θN , q ❞X j (q) - √ n Λ θN , τ s 0 gn θN , q λ (ϑ * n , q) ❞q. ✭✷✳✸✸✮ ◆♦✇ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ❛♥❞ t❤❡ ♣r♦❝❡ss❡s (X j (•), j = N + 1, . . . , n) ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✇✳r✳t✳ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss s 0 gn θN , q ❞X j (q) ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✼✶ ✐s ✇❡•• ❞❡✜♥❡❞✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤✐s st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✐s ❝❛••❡❞ ❙t✐❡•t•❡s ✐♥t❡❣r❛• ❛♥❞ ✐ts ❞❡✜♥✐t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ▲✐♣ts❡r ❛♥❞ ❙❤✐r❛②❡✈ ❬✹✽(✱ ❙❡❝t✐♦♥ ✶✽✳✹✳ ■♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✱ ✇❡ ❝❛♥ s❡❡ t❤❛t Bn θN , s = 1 √ n Λ θN , τ n j=N +1 s 0 hn θN , q ❞X j (q) - √ n Λ θN , τ s 0 hn θN , q λ (ϑ * n , q) ❞q ✭✷✳✸✹✮ ✐s t❤❡ ♠♦❞✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss B n (ϑ * n , s) = s 0 hn (ϑ * n , q) ❞U n (ϑ * n , q) . ❚❤❡r❡❢♦r❡✱ ✇❡ ✐♥tr♦❞✉❝❡ ∆ * n = τ 0 Ŵ * n (r) 2 Λ (ϑ * n , τ ) λ (ϑ * n , r) ❞r, ∆ ≡ 1 0 w 2 ν ❞ν, ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ❛♥❞ t❤❡ ❢♦r♠❛• ❡①♣r❡ss✐♦♥ ✭✷✳✷✾✮ ✐s r❡♣•❛❝❡❞ ❜② t❤❡ ❢♦••♦✇✐♥❣ st❛t✐st✐❝ Ŵ * n (r) = U n (ϑ * n , r) + 1 Λ( θN , τ ) r 0 φ+
2,n ( θN , s) φ1,n ( θN , s) Bn θN , s + ψ2,n ( θN , s) Ân θN , s λ θN , s ❞s.

❍❡r❡ t❤❡ ♣r♦❝❡ss❡s U n (•, •)✱ Ân (•, •) ❛♥❞ Bn (•, •) ❛r❡ ❣✐✈❡♥ ❜② ✭✷✳✶✷✮✱ ✭✷✳✸✸✮ ❛♥❞ ✭✷✳✸✹✮✱ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ ✜rst t❡st ✐s ❣✐✈❡♥ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❚❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✷✳✸✳✸✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ r❡❣✉•❛r✐t② R ❛♥❞ R 4 ❛r❡ ❢✉•✜••❡❞✱ t❤❡♥ t❤❡ t❡st ψ * n = ✶■ { ∆ * n >cα} , P ∆ > c α = α ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✳ Pr♦♦❢✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✐s t♦ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∆ * n =⇒ ∆ ✭✷✳✸✺✮ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✳ ✼✷ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❋♦r t❤❡ ✜rst t❡r♠ ♦❢ Ŵ * n (•)✱ ✇❡ ❛•r❡❛❞② ♣r♦✈❡❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ U n (ϑ * n , r) =⇒ U (ν), 0 ≤ ν ≤ 1, ✇❤❡r❡ U (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✶✺✮ ✭s❡❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✷✳✸✳✹✮✳ ❲❡ ♥♦✇ ❛♣♣•② ❚❛②•♦r✬s ❢♦r♠✉•❛ ❛♥❞ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ✱ t♦ ♦❜t❛✐♥ τ v Λ θN , z ❞z - τ v Λ (ϑ, z) ❞z ≤ θN -ϑ τ v Λ( θ, z) ❞z ≤ C θN -ϑ , ✇❤❡r❡ C ✐s s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ❣✐✈❡s t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✭❛s n → ∞) ✇✳r✳t✳ v ∈ [0, τ ] sup v∈[0,τ ] gn θN , v -g (ϑ, v) → 0. ❇② ✉s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts✱ ✇❡ ❤❛✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✇✳r✳t✳ v ∈ [0, τ ] sup v∈[0,τ ] hn θN , v -h (ϑ, v) → 0, sup v∈[0,τ ] J n ( θN ) -J(ϑ) → 0, sup v∈[0,τ ] ψ2,n ( θN , v) -ψ2 (ϑ, v) → 0, sup v∈[0,τ ] φ1,n ( θN , v) -φ1 (ϑ, v) → 0. ❚❤❡♥✱ ✇❡ ♣r♦❝❡❡❞ t♦ s❤♦✇ t❤❛t Ân θN , s =⇒  (ϑ, s) , ✇❤❡r❡ Ân (•, •) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✸✸✮ ❛♥❞  (ϑ, s) = s 0 g (ϑ, q) ❞u (q) = s 0 g (ϑ, q) ❞W (Λ (ϑ, q)) Λ (ϑ, τ ) - Λ (ϑ, τ ) J (ϑ) τ 0 τ q Λ (ϑ, z) ❞z ❞W (Λ (ϑ, q)) Λ (ϑ, τ ) s 0 g (ϑ, q) λ(ϑ, q) λ(ϑ, q) ❞Λ (ϑ, q) Λ (ϑ, τ ) . ❍❡r❡ t❤❡ ♣r♦❝❡ss u (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✷✵✮ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ g (ϑ, q) ❣✐✈❡♥ ❜② ✭✷✳✷✻✮ ✐s ❝♦♥t✐♥✉♦✉s ✇✳r✳t✳ q✳ ❇② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ t❤❡ ♣r♦❝❡ss ✭✷✳✸✸✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ Ân θN , s = 1 √ n Λ θN , τ n j=N +1 s 0 gn θN , q ❞π j (q) - √ n (ϑ * n -ϑ) 1 Λ θN , τ s 0 gn θN , q λ( θ, q) ❞q, ✇❤❡r❡ ❞π j (q) = ❞X j (q) -λ (ϑ, q) ❞q. ❯s✐♥❣ t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦✲ ❝❤❛st✐❝ ✐♥t❡❣r❛•s ❛♥❞ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❣✐✈❡♥ ❜② ✭✷✳✶✽✮✱ ✇❡ ♦❜t❛✐♥ ❞✐r❡❝t•② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ Ân θN , s =⇒  (ϑ, s) . ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✼✸ ◆♦✇ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s v = Λ (ϑ, q) Λ (ϑ, τ ) , r = Λ (ϑ, s) Λ (ϑ, τ ) , w(v) = W (Λ (ϑ, q)) Λ (ϑ, τ ) , ✭✷✳✸✻✮ ✐t ✐s ♦❜✈✐♦✉s t❤❛t  (ϑ, s) ≡ A (ϑ, r) , ✇❤❡r❡ A (ϑ, r) = r 0 g (ϑ, v) ❞w (v) - 1 0 g (ϑ, v) ❞w (v) r 0 g (ϑ, v) h (ϑ, v) ❞v. ❍❡r❡ t❤❡ ❢✉♥❝t✐♦♥s g (•, •) ❛♥❞ h (•, •) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✷✳✶✸✮ ❛♥❞ ✭✷✳✶✹✮✱ r❡s♣❡❝t✐✈❡•② ❛♥❞ w(•) ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❋✉rt❤❡r✱ ❜② t❤❡ s❛♠❡ ❛r❣✉♠❡♥ts✱ ✇❡ s❤♦✇ t❤❛t Bn θN , s =⇒ B(ϑ, s), ✇❤❡r❡ Bn (•, •) ✐s ❣✐✈❡♥ ❜② ✭✷✳✸✹✮ ❛♥❞ B (ϑ, s) = s 0 h (ϑ, q) ❞u (q) = s 0 h (ϑ, q) ❞W (Λ (ϑ, q)) Λ (ϑ, τ ) - Λ (ϑ, τ ) J (ϑ) τ 0 τ q Λ (ϑ, z) ❞z ❞W (Λ (ϑ, q)) Λ (ϑ, τ ) s 0 h (ϑ, q) λ(ϑ, q) λ(ϑ, q) ❞Λ (ϑ, q) Λ (ϑ, τ ) . ◆♦t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ h (ϑ, q) ❞❡✜♥❡❞ ❜② ✭✷✳✷✼✮ ✐s ❝♦♥t✐♥✉♦✉s ✇✳r✳t✳ q✳ ❯s✐♥❣ ❚❛②•♦r✬s ❢♦r♠✉•❛✱ ✐t ✐s ❡❛s✐•② s❡❡♥ t❤❛t Bn θN , s = 1 √ n Λ θN , τ n j=N +1 s 0 hn θN , q ❞π j (q) - √ n (ϑ * n -ϑ) 1 Λ θN , τ s 0 hn θN , q λ( θ, q) ❞q. ▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ ❜② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s ✭✷✳✸✻✮ B (ϑ, s) ≡ B (ϑ, r) , ✇❤❡r❡ B (ϑ, r) = r 0 h (ϑ, v) ❞w (v) - 1 0 g (ϑ, v) ❞w (v) r 0 h (ϑ, v) 2 ❞v. ❋✐♥❛••②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ✭✷✳✸✺✮ ❤♦•❞s✱ ❛♥❞ t❤❡ ❚❤❡♦r❡♠ ✐s ♣r♦✈❡❞✳ ✼✹ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✷✳✹ ❙❝♦r❡✲❢✉♥❝t✐♦♥ ❜❛s✐❝ st❛t✐st✐❝ ❖✉r ♦❜•❡❝t✐✈❡ ✐s t♦ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡st ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✳ ❚♦ ❞♦ t❤✐s✱ ✇❡ ✐♥tr♦❞✉❝❡ ✭❢♦r♠❛••②✮ t❤❡ s❝♦r❡✲❢✉♥❝t✐♦♥ ♣r♦❝❡ss ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✸(✮ u * n (t) = 1 √ n n j=1 t 0 λ (ϑ * n , v) λ (ϑ * n , v) [❞X j (v) -λ (ϑ * n , v) ❞v] , ✭✷✳✸✼✮ ✇❤✐❝❤ ❝♦♥✈❡r❣❡s t♦ u (•) ❞❡✜♥❡❞ ❜❡•♦✇ ❜② ✭✷✳✸✽✮✳ ❚❤❡♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r t❤❡ •✐♠✐t ♣r♦❝❡ss u (•) U (ν) = 1 √ τ ντ 0 λ (ϑ, v) λ (ϑ, v) ❞u(v), 0 ≤ ν ≤ 1 ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U (ν) = W (ν) - 1 0 g (ϑ, s) ❞W (s) ν 0 h (ϑ, s) ❞s. ❚❤❡ •❛st st❡♣ ✐s t♦ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L [•] ❢r♦♠ ❚❤❡♦r❡♠ ✶✳✹✳✶ ❛♥❞ t♦ ♦❜t❛✐♥ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L [U ] (ν) = w ν , 0 ≤ ν ≤ 1. ❇❡•♦✇ ✇❡ r❡❛•✐③❡ t❤✐s ♣r♦❣r❛♠ ✇✐t❤ t❤❡ ❡♠♣✐r✐❝❛• ♣r♦❝❡ss u * n (•)✱ ❛♣♣•② t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•] ❛♥❞ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ δn = 1 0 L[U * n ](ν) 2 ❞ν =⇒ 1 0 w 2 ν ❞ν. ❚❤❡r❡❢♦r❡ t❤❡ t❡st φn = ✶■ { δn>cα} ✇✐•• ❜❡ ❆❉❋ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢r❡❡ •✐♠✐t st❛t✐st✐❝✳ ■t ✐s ♦❜✈✐♦✉s t❤❛t t❤❡ ♣r♦❝❡ss ✭✷✳✸✼✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✭❢♦r✲ ♠❛••②✮ u * n (t) = 1 √ n n j=1 t 0 λ (ϑ * n , v) λ (ϑ * n , v) ❞X j (v) - 1 √ n n j=1 t 0 λ (ϑ * n , v) ❞v. ❍❡r❡ ✇❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤❡ ♣r♦❜•❡♠ ✐s ❛❜♦✉t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✇✳r✳t✳ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss (X j (•) , j = 1, . . . , n) t 0 λ(ϑ * n , v) λ(ϑ * n , v) ❞X j (v) ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✼✺ ❜❡❝❛✉s❡ t❤❡ ✐♥t❡❣r❛♥❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ❢✉t✉r❡✱ ✐✳❡✳✱ t❤❡ ▼❉❊ ϑ * n ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❤♦•❡ tr❛•❡❝t♦r② X n ✳ ❚❤❡ ❝♦rr❡❝t ❞❡✜♥✐t✐♦♥ ♦❢ t❤✐s ✐♥t❡❣r❛• ✇✐•• ❜❡ ❣✐✈❡♥ •❛t❡r✳ ❚❤❡ t❛s❦ ✐s ♥♦✇ t♦ ✜♥❞ ✭❛s ✇❡•• ❢♦r♠❛••②✮ t❤❡ •✐♠✐t ♦❢ t❤❡ ♣r♦❝❡ss u * n (•)✳ ❯s✐♥❣ ❚❛②•♦r✬s ❢♦r♠✉•❛✱ t❤❡ ❢♦r♠❛• ❡①♣❛♥s✐♦♥ ♦❢ u * n (•) ♣r♦✈✐❞❡s ✉s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥s ✿ u * n (t) = 1 √ n n j=1 t 0 λ (ϑ * n , v) λ (ϑ * n , v) [❞X j (v) -λ (ϑ, v) ❞v -(λ (ϑ * n , v) -λ (ϑ, v)) ❞v] = 1 √ n n j=1 t 0 λ(ϑ * n , v) λ(ϑ * n , v) ❞π j (v) - √ n (ϑ * n -ϑ) t 0 λ(ϑ * n , v) λ(ϑ * n , v) λ( θ, v) ❞v + o(1). ❚❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ✐♠♣•✐❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ 1 √ n n j=1 t 0 λ (ϑ, v) λ (ϑ, v) ❞π j (v) =⇒ t 0 λ (ϑ, v) λ (ϑ, v) ❞W v , ✇❤❡r❡ W v ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡♥✱ ❜② t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✳✾✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ u * n (t) =⇒ u(t) ≡ t 0 λ (ϑ, v) λ (ϑ, v) ❞W v -J (ϑ) -1 τ 0 τ v Λ (ϑ, s) ❞s λ (ϑ, v) ❞W v t 0 λ (ϑ, v) 2 λ (ϑ, v) ❞v. ✭✷✳✸✽✮ ■t ✐s ❡❛s✐•② s❡❡♥ t❤❛t t❤❡ •✐♠✐t ♣r♦❝❡ss u(•) ❞❡♣❡♥❞s ♦♥ Λ (•, •) ❛♥❞ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ❜❛s❡ ♦❢ t❤✐s st❛t✐st✐❝ ✐s ♥♦t ❆❉❋✳ ■♥❞❡❡❞ t♦ ♦❜t❛✐♥ t❤❡ ❆❉❋ •♦❋ t❡st ✇❡ ♣r♦♣♦s❡ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♣r♦❝❡ss ✭✷✳✸✽✮ ❛s ❢♦••♦✇s ✿ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ •❛✉ss✐❛♥ ♣r♦❝❡ss U (ν) = 1 √ τ ντ 0 λ (ϑ, v) λ (ϑ, v) ❞u(v), 0 ≤ ν ≤ 1 ✭✷✳✸✾✮ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s h (ϑ, s) = C 2 (ϑ) 1 2 J (ϑ) -1 λ (ϑ, sτ ) λ (ϑ, sτ ) ✭✷✳✹✵✮ ❛♥❞ g (ϑ, s) = C 2 (ϑ) -1 2 τ sτ Λ (ϑ, r) ❞r λ (ϑ, sτ ). ✭✷✳✹✶✮ ✼✻ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❍❡r❡ J (ϑ) = 1 0 λ (ϑ, sτ ) ❞s, 1 0 g (ϑ, s) 2 ❞s = 1 ❛♥❞ C 2 (ϑ) = 1 0 λ (ϑ, rτ ) τ rτ Λ (ϑ, z) ❞z 2 ❞r. ▲❡♠♠❛ ✷✳✹✳✺✳ ❲❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t② U (ν) = W (ν) - 1 0 g (ϑ, s) ❞W (s) ν 0 h (ϑ, s) ❞s, 0 ≤ ν ≤ 1, ✭✷✳✹✷✮ ✇❤❡r❡ W (ν) , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ Pr♦♦❢✳ ▲❡t ✉s ✜rst s❤♦✇ t❤❛t U (ν) = 1 √ τ ντ 0 λ (ϑ, v) λ (ϑ, v) ❞u(v) = 1 √ τ ντ 0 ❞W v - J (ϑ) -1 √ τ τ 0 τ v Λ (ϑ, z) ❞z λ (ϑ, v) ❞W v ντ 0 λ (ϑ, v) λ (ϑ, v) ❞v. ❇② t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s s = v τ ❛♥❞ W (ν) = τ -1/2 W ντ , 0 ≤ ν ≤ 1✱ ✇❡ ❝❛♥ ✇r✐t❡ U (ν) = 1 √ τ ν 0 ❞W sτ - J (ϑ) -1 √ τ 1 0 τ sτ Λ (ϑ, z) ❞z λ (ϑ, sτ ) ❞W sτ ν 0 λ (ϑ, sτ ) λ (ϑ, sτ ) ❞s = W (ν) - 1 0 g (ϑ, s) ❞W (s) ν 0 h (ϑ, s) ❞s. ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❝❡ss U (•) ❤❛s t❤❡ ❡①♣r❡ss✐♦♥ ✭✷✳✹✷✮ ❛♥❞ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛ ✐s ❝♦♠♣•❡t❡❞✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✇❡ ♦❜t❛✐♥ ♦♥❝❡ ♠♦r❡ t❤❡ s❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ •✐♠✐t ♣r♦❝❡ss ❣✐✈❡♥ ❜② ✭✷✳✶✺✮ ✐♥ ❙❡❝t✐♦♥ ✷✳✸✳ ❯s✐♥❣ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ U * n (•) ✭❞❡✜♥❡❞ ❜❡•♦✇ ❜② ✭✷✳✹✸✮✮ t♦ U (•) ✭s❡❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✹✳✹ ❜❡•♦✇✮ ❛♥❞ t❤❡ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ t❤❡♦r❡♠✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t δ n = 1 τ 1 0 ντ 0 λ (ϑ * n , v) λ (ϑ * n , v) ❞u * n (v) 2 ❞ν =⇒ δ ≡ 1 0 U (ν) 2 ❞ν ✐s s❛t✐s✜❡❞✳ ❖♥❡ ❝❛♥ s❡❡ t❤❛t t❤❡ t❡st ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ❜❛s❡ ♦❢ t❤✐s st❛t✐st✐❝ ψ * n = ✶ ■ {δn > cα} , P (δ > c α ) = α, α ∈ (0, 1) ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✼✼ ✐s ♥♦t ❆❉❋✳ ❚♦ ❛✈♦✐❞ t❤✐s ♣r♦❜•❡♠ ✇❡ ✇✐•• ❛♣♣•② t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•] ❞❡✜♥❡❞ ❜② ✭✶✳✷✻✮ t♦ t❤❡ ♣r♦❝❡ss ✭✷✳✹✷✮ t♦ ❣❡t t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss✱ ✐✳❡✱ 1 0 L[U ] (ν) 2 ❞ν = 1 0 w 2 ν ❞ν, 0 ≤ ν ≤ 1. ❖✉r ♥❡①t ❣♦❛• ✐s t♦ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡st ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t❤✐s ♣r♦♣❡rt② ✉s✐♥❣ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✇✐t❤ t❤❡ s❛♠❡ ❛❜♦✈❡ •✐♠✐t✳ ■♥❞❡❡❞ ✇❡ ❝❛•❝✉•❛t❡ ✭❢♦r♠❛••②✮ U * n (ν) = 1 √ τ ντ 0 λ (ϑ * n , v) λ (ϑ * n , v) ❞u * n (v), 0 ≤ ν ≤ 1, ✭✷✳✹✸✮ ✇❤❡r❡ t❤❡ ♣r♦❝❡ss u * n (•) ✐s ❣✐✈❡♥ ❜② ✭✷✳✸✼✮✳ ❋✉rt❤❡r✱ ✇❡ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L[•] t♦ t❤❡ ♣r♦❝❡ss U * n (•) ❛♥❞ ✇❡ s❤♦✇ t❤❛t t❤✐s st❛t✐st✐❝ ❝♦♥✈❡r❣❡s t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L[U * n ](ν) =⇒ w ν , 0 ≤ ν ≤ 1. ❚❤❡r❡❢♦r❡✱ t❤❡ t❡st φn = ✶ ■ { δn > cα} ✇✐t❤ ✭✉♥❞❡r ❤②♣♦t❤❡s✐s✮ δn = 1 0 L[U * n ](ν) 2 ❞ν =⇒ 1 0 w 2 ν ❞ν ✇✐•• ❜❡ ❆❉❋ ❜❡❝❛✉s❡ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ Λ (•, •) ❛♥❞ ϑ✳ ❚♦ ❞♦ t❤✐s •❡t ✉s ♣r♦❝❡❡❞ ❛s ❢♦••♦✇s ✿ ❲❡ ❤❛✈❡ ❢♦r t❤❡ st❛t✐st✐❝ U * n (•) ✭❢♦r♠❛••②✮ U * n (ν) = 1 √ τ ντ 0 λ (ϑ * n , v) λ (ϑ * n , v) ❞u * n (v) = 1 √ τ ντ 0 λ (ϑ * n , v) λ (ϑ * n , v) ❞ 1 √ n n j=1 v 0 λ(ϑ * n , s) λ(ϑ * n , s) [❞X j (s) -λ(ϑ * n , s) ❞s] = 1 √ n n j=1 1 √ τ ντ 0 1 λ(ϑ * n , v) ❞X j (v) - √ n √ τ ντ 0 λ(ϑ * n , v) ❞v. |❡♠❛r❦ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ντ 0 1 λ(ϑ * n , v) ❞X j (v) ✐s ♥♦t ✇❡•• ❞❡✜♥❡❞✳ ❚♦ s♦•✈❡ t❤✐s ❞✐✣❝✉•t② ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ❣✐✈❡♥ ❜② ✭✷✳✸✷✮ ❛♥❞ ❝♦♥str✉❝t❡❞ ❜② t❤❡ ♦❜s❡r✈❛t✐♦♥s X N = (X 1 , . . . , X N )✳ ❍❡♥❝❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ st❛t✐st✐❝ Ûn (ν) = 1 √ n n j=N +1 1 √ τ ντ 0 1 λ( θN , v) ❞X j (v) - √ n √ τ ντ 0 λ(ϑ * n , v) ❞v ✭✷✳✹✹✮ ✼✽ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❛♥❞ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✐s ✇❡•• ❞❡✜♥❡❞✳ ❇❡•♦✇ ✇❡ ✐♥s❡rt t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ✐♥ t❤❡ ❢✉♥❝t✐♦♥s h (•, •) ❛♥❞ g (•, •) ❞❡✜♥❡❞ ❜② ✭✷✳✹✵✮ ❛♥❞ ✭✷✳✹✶✮ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ h n (•, •) ❛♥❞ g n (•, •)✱ r❡s♣❡❝t✐✈❡•② ✿ h n θN , s = C 2 θN 1 2 Jn θN -1 λ θN , sτ λ θN , sτ ❛♥❞ g n θN , s = C 2 θN -1 2 τ sτ Λ θN , r ❞r λ θN , sτ . ❉❡♥♦t❡ I 1,n = r 0 g n θN , q 2 ❞q, I 2,n = r 0 h n θN , q g n θN , q ❞q, I 3,n = r 0 h n θN , q ❞q, I 4,n = r 0 h n θN , q 2 ❞q ❛♥❞ ♣✉t I 5,n = r 0 g n θN , q ❞q t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ ♦❢ ✭✶✳✷✷✮✳ ❚❤✐s ❛••♦✇s ✉s t♦ ✐♥tr♦❞✉❝❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡r✲ s✐♦♥s✑ ϕ 1,n (•, •), ϕ 2,n (•, •) ❛♥❞ ψ 2,n (•, •) ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ϕ 1 (•, •), ϕ 2 (•, •) ❛♥❞ ψ 2 (•, •) ❣✐✈❡♥ ❜② ✭✶✳✷✸✮✱ ✭✶✳✷✹✮ ❛♥❞ ✭✶✳✷✺✮✱ r❡s♣❡❝t✐✈❡•②✳ ❉✉❡ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ❛♥❞ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ✱ ✇❡ ❤❛✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✭❛s n → ∞) sup r∈[0,1) ϕ 2,n ( θN , r) -ϕ 2 (ϑ, r) -→ 0. ❲❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥ ϕ + 2,n θN , r =    ϕ 2,n θN , r -1 , ✐❢ ϕ 2,n θN , r > 0, 0, ❡•s❡, ✭✷✳✹✺✮ ✇❤✐❝❤ ❛s②♠♣t♦t✐❝❛••② ❝♦✐♥❝✐❞❡s ✇✐t❤ ϕ 2 (ϑ, r) -1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ •✐♠✐t ❞✐str✐❜✉✲ t✐♦♥ ❞♦❡s ♥♦t ❝❤❛♥❣❡✳ ◆❛♠❡•②✱ ❛s ❛ s✉✐t❛❜•❡ st❛t✐st✐❝ t♦ ❜❛s❡ t❤❡ t❡st✐♥❣ ♣r♦❝❡❞✉r❡ ♦♥ ✇❡ ✇✐•• t❛❦❡ δn = 1 0 W n (ν) 2 ❞ν, ∆ ≡ 1 0 w 2 ν ❞ν, ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✼✾ ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss ❛♥❞ W n (ν) = Ûn (ν) + ν 0 r 0 φ+ 2,n ( θN , r) φ1,n ( θN , r) h n θN , q + ψ2,n ( θN , r) g n θN , q ❞ Ûn (q) ❞r. ❍❡r❡ Ûn (•) ✐s ❞❡✜♥❡❞ ❜② ✭✷✳✹✹✮✳ ❚❤❡ s❡❝♦♥❞ t❡st ✐s ❣✐✈❡♥ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❚❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✷✳✹✳✹✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ r❡❣✉•❛r✐t② R ❛♥❞ R 4 ❛r❡ ❢✉•✜••❡❞✱ t❤❡♥ t❤❡ t❡st φn = ✶■ { δn>cα} , P ∆ > c α = α ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✳ Pr♦♦❢✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢ ✐s t♦ ✈❡r✐❢② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ δn =⇒ ∆ ✭✷✳✹✻✮ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✳ ◆♦t❡ t❤❛t ❞✉❡ t♦ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥ R 1 ❛♥❞ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ♣r❡•✐♠✐♥❛r② ❡st✐♠❛t♦r θN ✱ ✇❡ ❤❛✈❡ t❤❡ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡ ✭❛s n → ∞) ✇✳r✳t✳ s ∈ [0, 1] sup s∈[0,1] g n θN , s -g (ϑ, s) -→ 0, sup s∈[0,1] h n θN , s -h (ϑ, s) -→ 0, sup s∈[0,1] Jn ( θN ) -J(ϑ) → 0, sup s∈[0,1] ψ 2,n ( θN , s) -ψ 2 (ϑ, s) → 0 ❛♥❞ sup s∈[0,1] ϕ 1,n ( θN , s) -ϕ 1 (ϑ, s) → 0. ▼♦r❡♦✈❡r✱ t❤❡ ♣r♦❝❡ss Ûn (•) ❣✐✈❡♥ ❜② ✭✷✳✹✹✮ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ Ûn (ν) = 1 √ n n j=N +1 1 √ τ ντ 0 1 λ( θN , v) [❞X j (v) -λ(ϑ, v) ❞v] - √ n √ τ ντ 0 [λ(ϑ * n , v) -λ(ϑ, v)] λ( θN , v) ❞v = 1 √ n n j=N +1 1 √ τ ντ 0 1 λ( θN , v) ❞π j (v) - √ n (ϑ * n -ϑ) 1 √ τ ντ 0 λ( θ, v) λ( θN , v) ❞v + o(1), ✭✷✳✹✼✮ ✽✵ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✇❤❡r❡ π j (v) = X j (v) -Λ (ϑ, v) ✐s t❤❡ ❝❡♥t❡r❡❞ P♦✐ss♦♥ ♣r♦❝❡ss✳ ❇② t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s✱ ✇❡ ✜rst ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐✲ ❜✉t✐♦♥ 1 √ n n j=N +1 ντ 0 1 λ( θN , v) ❞π j (v) =⇒ ντ 0 ❞W v . ❚❤❡♥ ✉s✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✳✾✮✱ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s s = v τ ❛♥❞ W (ν) = τ -1/2 W ντ , 0 ≤ ν ≤ 1✱ ✇❡ ❣❡t t❤❡ s❡❝♦♥❞ ❝♦♥✈❡r❣❡♥❝❡ Ûn (ν) =⇒ 1 √ τ ντ 0 ❞W v - J (ϑ) -1 √ τ τ 0 τ v Λ (ϑ, z) ❞z λ (ϑ, v) ❞W v ντ 0 λ(ϑ, v) λ(ϑ, v) ❞v = W (ν) - 1 0 g (ϑ, s) ❞W (s) ν 0 h (ϑ, s) ❞s ≡ U (ν) , ✇❤❡r❡ W v , 0 ≤ v ≤ τ ❛♥❞ W (ν) , 0 ≤ ν ≤ 1 ❛r❡ s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss❡s✳ ◆♦✇ ✇❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ K n θN , r = r 0 g n θN , q ❞ Ûn (q) =⇒ K (ϑ, r) , ✇❤❡r❡ K (ϑ, r) = r 0 g (ϑ, s) ❞W (s) - 1 0 g (ϑ, s) ❞W (s) r 0 g (ϑ, s) h (ϑ, s) ❞s. h (ϑ, q) ❛♥❞ g (ϑ, q) ❞❡✜♥❡❞ ❜② ✭✷✳✹✵✮-✭✷✳✹✶✮ ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✇✳r✳t✳ q✳ ❇② ✭✷✳✹✼✮✱ ✐t ✐s ❡❛s✐•② s❡❡♥ t❤❛t K n (•, •) ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦••♦✇s ✿ K n θN , r = C 2 θN -1 2 √ τ √ n n j=N +1 r 0 τ qτ Λ( θN , z) ❞z ❞π j (qτ ) - √ n (ϑ * n -ϑ) C 2 θN -1 2 √ τ r 0 τ qτ Λ( θN , z) ❞z λ( θ, qτ ) ❞q = C 2 θN -1 2 √ τ √ n n j=N +1 rτ 0 τ v Λ( θN , z) ❞z ❞π j (v) - √ n (ϑ * n -ϑ) C 2 θN -1 2 √ τ rτ 0 τ v Λ( θN , z) ❞z λ( θ, v) ❞v, ✇❤❡r❡ ✇❡ ✉s❡ t❤❡ ❚❛②•♦r ❢♦r♠✉•❛ ❛♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ q = v τ ✳ ❇② t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ❛♥❞ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✳✾✮✱ ✇❡ ♦❜t❛✐♥ K n θN , r =⇒ C 2 (ϑ) -1 2 √ τ rτ 0 τ v Λ(ϑ, z) ❞z λ(ϑ, v) ❞W v ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ✽✶ - C 2 (ϑ) -1 2 √ τ J (ϑ) τ 0 τ v Λ (ϑ, z) ❞z λ (ϑ, v) ❞W v rτ 0 τ v Λ(ϑ, z) ❞z λ(ϑ, v) ❞v. ▲❡t ✉s ❝❤❛♥❣❡ t❤❡ ✈❛r✐❛❜•❡s s = v τ ❛♥❞ W (s) = τ -1 2 W sτ ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ K n θN , r =⇒ C 2 (ϑ) -1 2 r 0 τ sτ Λ(ϑ, z) ❞z λ(ϑ, sτ ) ❞W (s) - C 2 (ϑ) -1 2 J (ϑ) 1 0 τ sτ Λ (ϑ, z) ❞z λ (ϑ, sτ ) ❞W (s) r 0 τ sτ Λ(ϑ, z) ❞z λ(ϑ, sτ ) ❞s = K (ϑ, r) . ❙✐♠✐•❛r•②✱ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t L n θN , r = r 0 h n θN , q ❞ Ûn (q) L (ϑ, r) , ✇❤❡r❡ L (ϑ, r) = r 0 h (ϑ, s) ❞W (s) - 1 0 g (ϑ, s) ❞W (s) r 0 h (ϑ, s) 2 ❞s. ❯s✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ✭✷✳✹✼✮✱ ❚❛②•♦r✬s ❢♦r♠✉•❛ ❛♥❞ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡ q = v τ ✱ t❤❡ ♣r♦❝❡ss L n (•, •) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ L n θN , r = C 2 θN 1 2 Jn θN -1 √ τ √ n n j=N +1 r 0 λ( θN , qτ ) λ( θN , qτ ) ❞π j (qτ ) - √ n (ϑ * n -ϑ) C 2 θN 1 2 √ τ Jn θN -1 r 0 λ( θ, qτ ) λ( θN , qτ ) λ( θN , qτ ) ❞q = C 2 θN 1 2 Jn θN -1 √ τ √ n n j=N +1 rτ 0 λ( θN , v) λ( θN , v) ❞π j (v) - √ n (ϑ * n -ϑ) C 2 θN 1 2 Jn θN -1 √ τ rτ 0 λ( θ, v) λ( θN , v) λ( θN , v) ❞v. ❋✉rt❤❡r✱ ❜② t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ❛♥❞ t❤❡ r❡♣r❡s❡♥✲ t❛t✐♦♥ ✭✷✳✾✮✱ ✇❡ ♦❜t❛✐♥ L n θN , r =⇒ C 2 (ϑ) 1 2 J (ϑ) -1 √ τ rτ 0 λ(ϑ, v) λ(ϑ, v) ❞W v - C 2 (ϑ) 1 2 J (ϑ) -1 √ τ J (ϑ) τ 0 τ v Λ (ϑ, z) ❞z λ (ϑ, v) ❞W v rτ 0 λ(ϑ, v) 2 λ(ϑ, v) ❞v. ✽✷ ❈❍❆P❚❊| ✷✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ■◆❍❖▼❖•❊◆❊❖❯❙ P❖■❙❙❖◆ P|❖❈❊❙❙ ❚❤❡r❡❢♦r❡ ✉s✐♥❣ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜•❡s s = v τ ❛♥❞ W (s) = τ -1/2 W sτ , 0 ≤ s ≤ 1✱ ✇❡ ❝❛♥ ✇r✐t❡ L n θN , r =⇒ C 2 (ϑ) 1 2 J (ϑ) -1 r 0 λ(ϑ, sτ ) λ(ϑ, sτ ) ❞W (s) -C 2 (ϑ) 1 2 J (ϑ) -2 1 0 τ sτ Λ (ϑ, z) ❞z λ (ϑ, sτ ) ❞W (s) r 0 λ(ϑ, sτ ) 2 λ(ϑ, sτ ) ❞s = L (ϑ, r) . ❋✐♥❛••②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✷✳✹✻✮ ✐s ♣r♦✈❡❞ ❛♥❞ ✉s✐♥❣ t❤✐s r❡s✉•t t❤❡ t❡st φn ✐s ❆❉❋✳ ❈❤❛♣t❡r ✸ ❖♥ •♦❋ ❚❡st ❢♦r ❊r❣♦❞✐❝ ❉✐✛✉s✐♦♥ Pr♦❝❡ss ✸✳✶ ■♥tr♦❞✉❝t✐♦♥ ❚❤✐s ❝❤❛♣t❡r ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ♣r♦❜•❡♠ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •♦❋ t❡st ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ❜❛s✐❝ ❤②♣♦t❤❡s✐s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✳ ❲❡ ♣r♦♣♦s❡ ❛♥ ❆❉❋ t❡st✱ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t✇♦ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ❛♣♣•✐❡❞ t♦ t❤❡ ♥♦r♠❛•✐③❡❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❡♠♣✐r✐❝❛• ❞❡♥s✐t②✳ |❡♠✐♥❞ ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ s✐♠♣•❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s ✭s❡❡ ❬✹✶( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮ ✿ H 0 ✿ t❤❡ ♦❜s❡r✈❡❞ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss s❛t✐s✜❡s t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❞X t = S 0 (X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ✇❤❡r❡ W t ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✱ S 0 (•) ✐s s♦♠❡ ❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❛♥❞ X 0 ✐s t❤❡ ✐♥✐t✐❛• ✈❛•✉❡ ♦❢ X t ✳ ■♥ t❤✐s st❛t❡♠❡♥t✱ t❤❡ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❡r❣♦❞✐❝ ✇✐t❤ t❤❡ ❞❡♥s✐t② ♦❢ ✐♥✈❛r✐❛♥t •❛✇ f S 0 (x)✳ ❲❡ ❞❡♥♦t❡ ❜② F S 0 (x) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐str✐✲ ❜✉t✐♦♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛• ❞❡♥s✐t② fT (x) ✭✇✐•• ❜❡ ❞❡✜♥❡❞ ❜② ✭✸✳✾✮ ❜❡•♦✇✮ ❣✐✈❡♥ ❜② δT = ∞ -∞ [ζ T (x)] 2 ❞F S 0 (x) , ζ T (x) = √ T fT (x) -f S 0 (x) ✽✸ ✽✹ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ •✐♠✐t ✭✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s H 0 ✮ δ (S 0 ) = ∞ -∞ ζ (S 0 , x) 2 ❞F S 0 (x) . ❍❡r❡ ζ (S 0 , x) = 2 f S 0 (x) ∞ -∞ F S 0 (y) -✶■ {y>x} σ(y) f S 0 (y) ❞W (y) , ✇❤❡r❡ W (•) ✐s ❞♦✉❜•❡✲s✐❞❡❞ ❲✐❡♥❡r ♣r♦❝❡ss ❛♥❞ t❤❡ •✐♠✐t ♣r♦❝❡ss δ (S 0 ) ❞❡♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡• ✭t❤❡ ♣r♦♦❢ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✸✾(✮✳ ❚♦ ❛✈♦✐❞ t❤✐s ♣r♦❜•❡♠✱ ❛ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L 1 [ζ T ] (•) ♦❢ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ ζ T (•) ✇❛s ♣r♦♣♦s❡❞ ✭s❡❡ ❬✹✶( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮ s✉❝❤ t❤❛t δT = ∞ -∞ L 1 [ζ T ] (x) 2 ❞F S 0 (x) =⇒ 1 0 w 2 s ❞s, ✭✸✳✶✮ ✇❤❡r❡ w s , 0 ≤ s ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡♥✱ ❛❢t❡r tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡ t❡st st❛t✐st✐❝ ✐s ❛s ❢♦••♦✇s ✿ δ * T = ∞ -∞ 1 √ T T 0 ✶■ {Xs<x} σ (X s ) [❞X s -S 0 (X s ) ❞s] 2 ❞F S 0 (x) ✇✐t❤ t❤❡ s❛♠❡ •✐♠✐t ✭✸✳✶✮✳ ◆♦t❡ t❤❛t t❤✐s st❛t✐st✐❝ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ t♦ ❝♦♥str✉❝t t❤❡ ❑♦•♠♦❣♦r♦✈✲❙♠✐r♥♦✈ t②♣❡ ❆❉❋ t❡st ✐♥ ❬✺✶(✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t❡st φ * T = ✶■ { δ * T >cα} , P 1 0 w 2 s ❞s > c α = α ✐s ❆❉❋ ✉♥❞❡r t❤❡ s✐♠♣•❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s H 0 ✳ ❚❤✐s ♣r♦❜•❡♠ ✇❛s ❛•s♦ st✉❞✐❡❞ ✐♥ ❬✸✹( ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 t❤❛t ❞X t = S(ϑ, X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ϑ ∈ Θ = (a, b), ✐✳❡✳✱ t❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t S (ϑ, x) ✐s s♦♠❡ ❦♥♦✇♥ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② fT (x) t❤❡ •♦❝❛• t✐♠❡ ❡st✐♠❛t♦r ✭❡♠♣✐r✐❝❛• ❞❡♥s✐t②✮ ♦❢ t❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② f (ϑ, x) ✇❤✐❝❤ ✇✐•• ❜❡ ❞❡✜♥❡❞ ❜② ✭✸✳✾✮ ❛♥❞ ✭✸✳✽✮ ❜❡•♦✇✱ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡ st❛t✐st✐❝ ✐s ❣✐✈❡♥ ❜② ∆T = ∞ -∞ μT ( θT , x) 2 ❞F ( θT , x), μT ( θT , x) = √ T fT (x) -f ( θT , x) , ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✽✺ ✇❤❡r❡ θT ✐s t❤❡ ▼▲❊ ♦❢ t❤❡ ♣❛r❛♠❡t❡r ϑ✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤❡ ✐♠♠❡❞✐❛t❡ ✉s❡ ♦❢ t❤❡ t❡st φT = ✶■ { ∆T >dα} •❡❛❞s t♦ t❤❡ s❛♠❡ ♣r♦❜•❡♠ ❛s ✐♥ t❤❡ ✐✳✐✳❞✳ ❝❛s❡✳ ■♥❞❡❡❞✱ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤✐s st❛t✐st✐❝ ✭❛s T → ∞) ✉♥❞❡r H 0 ❞❡✲ ♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡•✱ ✐✳❡✱ S (•, •) , σ (•) ❛♥❞ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❚♦ s♦•✈❡ t❤✐s ♣r♦❜•❡♠✱ ❛ ✜rst •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L 1 [μ T ] (•) ♦❢ t❤❡ st❛t✐st✐❝ μT (•, •) ✇❛s ♣r♦♣♦s❡❞ ✐♥ ❬✸✹( ❛♥❞ ❬✹✶(✳ ▼♦r❡♦✈❡r✱ ✐t ✇❛s s❤♦✇♥ t❤❛t t❤✐s •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛s②♠♣t♦t✐❝❛••② ❡q✉✐✈❛•❡♥t t♦ t❤❡ st❛t✐st✐❝ ξT ( θT , x) = 1 √ T T 0 ✶■ {Xs<x} σ (X s ) ❞X s -S( θT , X s ) ❞s ✇✐t❤ t❤❡ s❛♠❡ •✐♠✐t ✭❛s T → ∞✮ U (t) = W (t) - 1 0 h (ϑ, s) dW (s) t 0 h (ϑ, s) ds, 0 ≤ t ≤ 1. ✭✸✳✷✮ ❍❡r❡ W (s), 0 ≤ s ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡ •✐♠✐t ♣r♦❝❡ss U (•) ✐s ♥♦t ❞✐s✲ tr✐❜✉t✐♦♥ ❢r❡❡✳ ❍❡♥❝❡✱ t❤❡ ❆❉❋ t❡st ✇❛s ❜❛s❡❞ ♦♥ ❛ s❡❝♦♥❞ •✐♥❡❛r tr❛♥s❢♦r♠❛✲ t✐♦♥ L 2 [•] ✭❝❛♥ ❜❡ ❢♦✉♥❞✱ ❡✳❣✳✱ ✐♥ ❬✸✵( ❛♥❞ ❬✸✹(✮ ♦❢ t❤❡ st❛t✐st✐❝ ξT (•, •) s✉❝❤ t❤❛t L 2 [U ] (t) = w t ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ∞ -∞ L 2 ξT ( θT , •) (x) 2 ❞F ( θT , x) =⇒ 1 0 w 2 t ❞t ✇❛s ♣r♦✈❡❞✳ ❍❡r❡ w t , 0 ≤ t ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛• •♦❋ t❡sts ❢♦r t❤❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❞✐✛✉s✐♦♥ ❛♥❞ ✐♥❤♦♠♦✲ ❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ♣r♦♣♦s❡❞✱ ❢♦r ❡①❛♠♣•❡✱ ✐♥ t❤❡ ✇♦r❦s ❬✺✻(✱ ❬✹✹(✱ ❬✹✷( ❛♥❞ ❬✶✶(✳ ❙❡❡ ❛•s♦ ❬✶✷( ✇❤❡r❡ s♦♠❡ •♦❋ t❡sts ❢♦r ❞✐✛✉s✐♦♥ ❛♥❞ ✐♥❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss❡s ✇✐t❤ s✐♠♣•❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s ✇❡r❡ ♣r♦♣♦s❡❞✳ ■t ✇❛s s❤♦✇♥ t❤❛t t❤❡s❡ t❡sts ❛r❡ ❆❉❋✳ ❚❤❡ ❑♦•♠♦❣♦r♦✈✲❙♠✐r♥♦✈ st❛t✐st✐❝ ❢♦r ❡r❣♦❞✐❝ ❞✐✛✉✲ s✐♦♥ ♣r♦❝❡ss ✇❛s st✉❞✐❡❞✱ ❡✳❣✳✱ ✐♥ ❬✷✵( ❛♥❞ ❬✷✶(✳ ■♥ ❬✸✾(✱ ❙❡❝t✐♦♥ ✺✳✹✱ t❤❡ ❛✉t❤♦r ❞✐s❝✉ss❡s s♦♠❡ ♣♦ss✐❜✐•✐t✐❡s ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❑♦•♠♦❣♦r♦✈✲❙♠✐r♥♦✈ ❛♥❞ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t❡sts ✇❤❡♥ t❤❡ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✐s ♦❜s❡r✈❡❞ ♦♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❚❤❡ s❛♠❡ ♣r♦❜•❡♠ ✇❛s ❝♦♥s✐❞❡r❡❞ ✐♥ ❬✷✸( ❜✉t ❢♦r t❤❡ ❤②♣♦✲ t❤❡s✐s ✇✐t❤ s✐❣♥✲t②♣❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t✳ ■♥ t❤✐s ❝❛s❡✱ t✇♦ ❈r❛♠ér✲✈♦♥ ▼✐s❡s •♦❋ t❡sts ❜❛s❡❞ ♦♥ ❡♠♣✐r✐❝❛• ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❡♠♣✐r✐❝❛• ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✇❡r❡ st✉❞✐❡❞✳ ▼♦r❡ ❛❜♦✉t •♦❋ t❡sts ❢♦r t❤❡ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ ❢♦✉♥❞✱ ✽✻ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❡✳❣✳✱ ✐♥ ❑•❡♣ts②♥❛ & ❑✉t♦②❛♥ts ❬✸✹( ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ❤②♣♦t❤❡s❡s ❛♥❞ ◆❡❣r✐ & ◆✐s❤✐②❛♠❛ ❬✺✶( ❢♦r t❤❡ s✐♠♣•❡ ❤②♣♦t❤❡s❡s✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤❡ ❣❡♥❡r❛• ❝❛s❡ ♦❢ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✇✐t❤ ✉♥❦✲ ♥♦✇♥ s❤✐❢t ✭♦♥❡ ❞✐♠❡♥s✐♦♥❛•✮ ♣❛r❛♠❡t❡r ✇❛s st✉❞✐❡❞ ✐♥ ❬✺✹(✳ ❚❤❡ ❛✉t❤♦rs s❤♦✇❡❞ t❤❛t t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ❛♥❞ t❤❡r❡❢♦r❡ ✐s ❛s②♠♣t♦t✐❝❛••② ♣❛r❛♠❡t❡r ❢r❡❡ ✭❆P❋✮✳ ▼♦r❡♦✈❡r✱ t❤❡ ❆P❋ ❑♦•♠♦❣♦r♦✈✲❙♠✐r♥♦✈ t②♣❡ t❡sts ✇❡r❡ st✉❞✐❡❞ ✐♥ ❬✻✷(✳ ❙✐♠✐•❛r r❡s✉•ts ❢♦r t❤❡ ❆P❋ t❡sts ♦❢ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❜② ❑✉t♦②❛♥ts ❬✹✶(✳ ❚❤❡ ❛✉t❤♦r st✉❞✐❡❞ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t❡ ❜❛s✐❝ ❤②♣♦t❤❡s✐s ✇✐t❤ ❛ ♣❛r❛♠❡tr✐❝ ❝•❛ss ♦❢ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s ✐♥❝•✉❞✐♥❣ t❤❡ ❖r♥st❡✐♥✲❯❤•❡♥❜❡❝❦ ❛♥❞ s✐♠♣•❡ s✇✐t❝❤✐♥❣ ♣r♦❝❡ss❡s✳ ▼♦r❡ ♣r♦❜•❡♠s ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •♦❋ t❡sts ❢♦r t❤❡ ❞✐s❝r❡t❡ t✐♠❡ ♦❜s❡r✈❛t✐♦♥s ✇❡r❡ ✇✐❞❡•② st✉❞✐❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✱ ❡✳❣✳✱ ◆❡❣r✐ ✫ ◆✐s❤✐②❛♠❛ ❬✺✸( ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ s♠❛•• ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✳ ■♥ ❬✺✷(✱ t❤❡ ♣r♦♣♦s❡❞ ❆❉❋ •♦❋ t❡sts ✇❡r❡ ❜❛s❡❞ ♦♥ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❛t✐♦♥s✱ ♦♥ ❞✐s❝r❡t❡ t✐♠❡ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ♦♥ t❤❡ s♦✲❝❛••❡❞ t✐❝❦ t✐♠❡ s❛♠♣•❡❞ ❞❛t❛ ❢♦r ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s✳ ❲❡ ❤❛✈❡ t♦ ❡♠♣❤❛s✐③❡ t❤❛t ✐♥ ❬✸✹( ❛♥❞ ✐♥ ♠❛♥② ♦t❤❡r ✇♦r❦s✱ ❡✳❣✳✱ ❬✹✾( t❤❡ ❡st✐♠❛t♦r ✉s❡❞ ✇❛s ❛•✇❛②s t❤❡ ▼▲❊ ❛♥❞ t❤✐s ✐s ✐♠♣♦rt❛♥t ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ■♥ ❛•• s✉❝❤ ❝❛s❡s✱ t❤❡ •✐♠✐t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ✉♥❞❡r•②✐♥❣ st❛t✐st✐❝s ✇❛s ❛•✇❛②s •✐❦❡ ✭✸✳✷✮✳ ❍♦✇❡✈❡r✱ ✐♥ s♦♠❡ ♣r♦❜•❡♠s t❤❡ ▼▲❊ ✐s ♥♦t ❣✐✈❡♥ ✐♥ ❛♥ ❡①♣•✐❝✐t ❢♦r♠ ❛♥❞ t❤❡r❡❢♦r❡ s♦♠❡t✐♠❡s ✐t ✐s ❜❡tt❡r t♦ ✉s❡ ♦t❤❡r ❡st✐♠❛t♦rs •✐❦❡ t❤❡ ▼❉❊✳ ▲❛t❡r✱ ✇❡ ✇✐•• ❣❡t t❤❡ •✐♠✐t ♣r♦❝❡ss ✭✸✳✸✮ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s h (•, •) ❛♥❞ g (•, •) ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ▼❉❊✳ ❚❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ❝❤❛♣t❡r ✐s t♦ ❝♦♥str✉❝t ❛♥ ❆❉❋ t❡st ❜❛s❡❞ ♦♥ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✶✳✷✻✮ ✇❤❡♥ t❤❡ ❜❛s✐❝ ♠♦❞❡• ✐s ❛ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❡r❣♦✲ ❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss X T = {X t , 0 ≤ t ≤ T } s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❞X t = S(X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T. ❍❡r❡ W t , 0 ≤ t ≤ T ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t S (•) ✐s ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✱ t❤❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t σ (x) 2 > 0 ✐s ❦♥♦✇♥ ❛♥❞ X 0 ✐s t❤❡ ✐♥✐t✐❛• ✈❛•✉❡ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✽✼ ♦❢ X t ✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s✐s H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ✐✳❡✱ t❤❡ ♣r♦❝❡ss X T ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ ❡q✉❛t✐♦♥ ✿ ❞X t = S(ϑ, X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ✇❤❡r❡ S (ϑ, x) ✐s ❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss X T ✐s r❡❝✉rr❡♥t ♣♦s✐t✐✈❡✱ ✐✳❡✳✱ ✐t ❤❛s ❡r❣♦❞✐❝ ♣r♦♣❡rt✐❡s ✐♥ ❙❡❝t✐♦♥ ✸✳✷✳ ■♥ ❙❡❝t✐♦♥ ✸✳✸✱ ✇❡ ♣r♦♣♦s❡ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ t❤❡ ▼❉❊ ϑ * T = arg inf ϑ∈Θ R fT (x) -f (ϑ, x) 2 ❞x.
❲❡ ❝♦♥s✐❞❡r t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝

δ T = ∞ -∞ η * T (ϑ * T , x) 2 ❞F (ϑ * T , x) ❛♥❞ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ η * T (ϑ * T , x) = √ T fT (x) -f (ϑ * T , x) =⇒ η (ϑ, x) = 2 f (ϑ, x) +∞ -∞ F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) ❞W (y) -2 J (ϑ) -1 R R F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) f (ϑ, x) ḟ (ϑ, x) ❞x ❞W (y) ḟ (ϑ, x). ❆ ✜rst •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✹✶(✮ L 1 [η] (x) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) = U (F (ϑ, x)) ✇✐•• ❜❡ ❛♣♣•✐❡❞ ❢♦r t❤❡ •✐♠✐t ♣r♦❝❡ss η (•, •)✳ ❚❤❡♥ ✇❡ s❤♦✇ t❤❛t t❤❡ ❛❜♦✈❡ tr❛♥s✲ ❢♦r♠❛t✐♦♥ ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U (t) = w (t) - 1 0 g (ϑ, s) ❞w (s) t 0 h (ϑ, s) ❞s, 0 ≤ t ≤ 1, ✭✸✳✸✮ ✇❤❡r❡ w (t) , 0 ≤ t ≤ 1 ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡ ♥❡①t st❡♣ ✐s t♦ ✐♥tr♦❞✉❝❡ t❤❡ s❡❝♦♥❞ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L 2 [•] ✭✶✳✷✻✮ ❛♥❞ t♦ ♦❜t❛✐♥ L 2 [U ] (t) = w t , 0 ≤ t ≤ 1. ✭✸✳✹✮ ✽✽ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ◆♦✇✱ ✇❡ ❤❛✈❡ t♦ r❡❛•✐③❡ t❤❡ s✐♠✐•❛r tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ♣r♦❝❡ss ξ T (•, •) ❛♥❞ t❤❡♥ ✇❡ s❤♦✇ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L 2 [ξ T ] (x) =⇒ L 2 [U ] (t) = w t , 0 ≤ t ≤ 1. ◆♦t❡ t❤❛t t❤❡ ♣r♦❝❡ss ξ T (•, •) ❞❡✜♥❡❞ ❜② ✭✸✳✷✷✮ ❜❡•♦✇ ✐s ❛s②♠♣t♦t✐❝❛••② ❡q✉✐✈❛•❡♥t t♦ L 1 [η * T ] (•)✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ δ * T = ∞ -∞ L 2 [ξ T ] (x) 2 dF (ϑ * T , x) =⇒ 1 0 w 2 t dt. ❚❤❡r❡❢♦r❡✱ t❤❡ t❡st ψ * T = ✶■ {δ * T >cα} ✇✐•• ❜❡ ❆❉❋ ❜❡❝❛✉s❡ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ δ * T ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ S (•, •)✱ σ (•) ❛♥❞ ϑ✳ ✸✳✷ Pr❡•✐♠✐♥❛r✐❡s ❙✉♣♣♦s❡ t❤❛t ✇❡ ❤❛✈❡ ❝♦♥t✐♥✉♦✉s t✐♠❡ ♦❜s❡r✈❛t✐♦♥s X T = {X t , 0 ≤ t ≤ T } ♦❢ ❛ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✱ ✇❤✐❝❤ ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ❞X t = S(X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T, ✭✸✳✺✮ ✇❤❡r❡ W t , 0 ≤ t ≤ T ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡ tr❡♥❞ ❝♦❡✣❝✐❡♥t S (•) ✐s ❛♥ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥✱ t❤❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t σ (x) 2 > 0 ✐s ❦♥♦✇♥ ❛♥❞ X 0 ✐s t❤❡ ✐♥✐t✐❛• ✈❛•✉❡ ♦❢ X t ✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜•❡♠ ♦❢ t❡st✐♥❣ t❤❡ ♣❛r❛♠❡tr✐❝ ✭❝♦♠♣♦s✐t❡✮ ❤②♣♦t❤❡s✐s H 0 : S(•) ∈ {S(ϑ, •), ϑ ∈ Θ = (a, b)} , ✇❤❡r❡ 0 < a < b < ∞, ✐✳❡✳✱ t❤❡ ♣r♦❝❡ss X T ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❞X t = S(ϑ, X t ) ❞t + σ (X t ) ❞W t , X 0 , 0 ≤ t ≤ T. ✭✸✳✻✮ ❍❡r❡ S (ϑ, x) ✐s ❦♥♦✇♥ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❲❡ s✉♣♣♦s❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s S (•) ❛♥❞ σ (•) ♦❢ t❤❡ ♦❜s❡r✈❡❞ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥s ES ✭❡①✐st❡♥❝❡ ♦❢ t❤❡ s♦•✉t✐♦♥✮✱ A 0 ❛♥❞ RP ✭r❡❝✉rr❡♥t ♣♦s✐t✐✈❡✮✭❡r❣♦❞✐❝✐t②✮ ✿ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✽✾ ES. ❚❤❡ ❢✉♥❝t✐♦♥ S (•, •) ✐s •♦❝❛••② ❜♦✉♥❞❡❞✱ t❤❡ ❢✉♥❝t✐♦♥ σ (•) 2 ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ♣♦s✐t✐✈❡ ❛♥❞ ❢♦r s♦♠❡ C > 0, t❤❡ ❝♦♥❞✐t✐♦♥ xS (ϑ, x) + σ (x) 2 ≤ C 1 + x 2 ❤♦•❞s✳ ❇② t❤✐s ❝♦♥❞✐t✐♦♥✱ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛• ❡q✉❛t✐♦♥ ✭✸✳✺✮ ❤❛s ❛ ✉♥✐q✉❡ ✇❡❛❦ s♦•✉t✐♦♥ ❢♦r ❛•• ϑ ∈ Θ ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬✶✾(✮✳ ❉❡♥♦t❡ ❜② P t❤❡ ❝•❛ss ♦❢ •♦❝❛••② ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✇✐t❤ ♣♦•②♥♦♠✐❛• ♠❛•♦r❛♥ts (p > 0) P = {h (•) : |h(y)| ≤ C (1 + |y| p )} . ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ ❢♦••♦✇✐♥❣ t✇♦ ❝♦♥❞✐t✐♦♥s ✭s❡❡ ❬✸✾( ❢♦r ♠♦r❡ ❞❡t❛✐•s✮ ✿ A 0 . ❚❤❡ ❢✉♥❝t✐♦♥s S (•, •)✱ σ (•) ±1 ∈ P ❛♥❞ ❢♦r ❛•• ϑ lim |y|→∞ sup ϑ∈Θ s❣♥ (y) S (ϑ, y) σ (y) 2 < 0. RP. ◆♦t❡ t❤❛t ✐❢ S (•, •) ❛♥❞ σ (•) s❛t✐s❢② A 0 ✱ t❤❡♥ ✇❡ ❤❛✈❡ V (ϑ, x) = x 0 exp -2 y 0 S (ϑ, z) σ (z) 2 ❞z ❞y -→ ±∞, ❛s x → ±∞ ❛♥❞ G (ϑ) = +∞ -∞ σ (x) 2 exp 2 x 0 S (ϑ, y) σ (y) 2 ❞y ❞x < ∞. ✭✸✳✼✮ ■❢ ❜♦t❤ ❝♦♥❞✐t✐♦♥s ❛r❡ ❢✉•✜••❡❞ t❤❡♥ t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss X T ✐s r❡❝✉rr❡♥t ♣♦✲ s✐t✐✈❡✱ ✐✳❡✳✱ ✐t ❤❛s ❡r❣♦❞✐❝ ♣r♦♣❡rt✐❡s✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭s❡❡✱ ❡✳❣✳✱ ❬✸✾(✱ ❚❤❡♦r❡♠ ✶✳✶✻✮ f (ϑ, x) = 1 G (ϑ) σ (x) 2 exp 2 x 0 S (ϑ, y) σ (y) 2 ❞y , x ∈ R, ✭✸✳✽✮ ✇❤❡r❡ G (ϑ) ❞❡✜♥❡❞ ❜② ✭✸✳✼✮ ✐s t❤❡ ♥♦r♠❛•✐③✐♥❣ ❝♦♥st❛♥t✳ ❲❡ ❞❡♥♦t❡ ❜② F (ϑ, x) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥✳ ■♥tr♦❞✉❝❡ t❤❡ r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R✳ ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❑✉t♦②❛♥ts ❬✸✾(✮ R 1 ✳ ❚❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② ❢✉♥❝t✐♦♥ f (ϑ, x) ✐s ❞✐✛❡r❡♥t✐❛❜•❡ ✇✳r✳t✳ ϑ ❢♦r ❛•• x ∈ R ❛♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ḟ (ϑ, x) ∈ P ✐s ✉♥✐❢♦r♠•② ❝♦♥t✐♥✉♦✉s ✐♥ t❤❡ ❢♦••♦✇✐♥❣ s❡♥s❡ ✿ ❢♦r ❛♥② ϑ 0 ∈ Θ ❛♥❞ ν > 0 lim ν→0 sup ϑ 0 ∈Θ sup |ϑ-ϑ 0 |≤ν ḟ (ϑ, •) -ḟ (ϑ 0 , •) = 0. ✾✵ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ R 2 ✳ ❚❤❡ ✐❞❡♥t✐✜❛❜✐•✐t② ❝♦♥❞✐t✐♦♥ ✐s inf |ϑ-ϑ 0 |>ν f (ϑ, •) -f (ϑ 0 , •) > 0. R 3 ✳ ❚❤❡ ❢✉♥❝t✐♦♥ J (ϑ) = ḟ (ϑ, •) 2 ✐s ♣♦s✐t✐✈❡ ✉♥✐❢♦r♠•② ✐♥ ϑ✳ ❍❡r❡ • ✐s t❤❡ L 2 ✲♥♦r♠ ✿ ḟ (ϑ, •) 2 = R ḟ (ϑ, x) 2 ❞x. ■♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❜❡•♦✇ ✇❡ s✉♣♣♦s❡ t❤❛t t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ ♥✉•• ❤②♣♦✲ t❤❡s✐s H 0 ❛r❡ ❛•✇❛②s ❢✉•✜••❡❞✳ ✸✳✸ ▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r ❚❤❡ ▼❉❊ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ❤❡•♣ ♦❢ t❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭✸✳✽✮ ❛♥❞ t❤❡ ❡♠♣✐r✐❝❛• ❞❡♥s✐t② fT (x) ❞❡✜♥❡❞ ❜② t❤❡ r❡•❛t✐♦♥ fT (x) = 2Λ T (x) σ(x) 2 T , ✭✸✳✾✮ ✇❤❡r❡ Λ T (x) ✐s t❤❡ •♦❝❛• t✐♠❡ ♦❢ t❤❡ ♦❜s❡r✈❡❞ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✭s❡❡✱ ❡✳❣✳✱ ❬✸✾( ❈❤❛♣t❡r ✶✱ ❙❡❝t✐♦♥ ✶✳✶✳✸ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥✮✳ ❚❤❡ ▼❉❊ ✐s ❞❡✜♥❡❞ ❛s ❛ s♦•✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜•❡♠ fT (•) -f (ϑ * T , •) = inf ϑ∈Θ fT (•) -f (ϑ, •) , ✭✸✳✶✵✮ ✇❤❡r❡ ϑ ∈ Θ = (a, b) ✐s t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ✭✉♥❦♥♦✇♥✮ ♣❛r❛♠❡t❡r✳ ◆♦t❡ t❤❛t t❤❡ ♣r♦♣❡rt✐❡s ❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ (T → ∞) ❜❡❤❛✈✐♦r ♦❢ t❤❡ ▼❉❊ ❢♦r t❤✐s ♠♦❞❡• ✇❡r❡ st✉❞✐❡❞ ✉♥❞❡r r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ✐♥ ❬✸✾(✳ ❚❤❡ ▼❉❊ ϑ * T ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ✇❡•• ❛s ❢♦••♦✇s ✿ ϑ * T = arg inf ϑ∈Θ R fT (x) -f (ϑ, x) 2 ❞x ✭✸✳✶✶✮ ❛♥❞ ✐t s❛t✐s✜❡s t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡q✉❛t✐♦♥ ✭▼❉❊q✮ ✭❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✸✾(✮ R fT (x) -f (ϑ * T , x) ḟ (ϑ * T , x) ❞x = 0. ▲❡t ✉s ♣✉t u * T = √ T (ϑ * T -ϑ)✳ ❚❤❡♥✱ ❜② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ f (ϑ * T , x) = f (ϑ, x) + (ϑ * T -ϑ) ḟ ( θ, x), ✇❤❡r❡ | θ -ϑ| ≤ |ϑ * T -ϑ|✱ t❤❡ ▼❉❊q ❜❡❝♦♠❡s ❛s ❢♦••♦✇s ✿ R √ T fT (x) -f (ϑ, x) -u * T ḟ ( θ, x) ḟ (ϑ * T , x) ❞x = 0. ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✾✶ ❯s✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ▼❉❊✱ ✇❡ ❝❛♥ ✇r✐t❡ u * T = R ḟ (ϑ, x) 2 ❞x -1 R √ T fT (x) -f (ϑ, x) ḟ (ϑ, x) ❞x + o(1). |❡♠✐♥❞ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ √ T fT (x) -f (ϑ, x) =⇒ 2 f (ϑ, x) ∞ -∞ F (ϑ, y) -✶ ■ {y>x} σ(y) f (ϑ, y) ❞W (y) , ✭✸✳✶✷✮ ✇❤❡r❡ W (•) ✐s ❞♦✉❜•❡✲s✐❞❡❞ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❋♦r t❤❡ ♣r♦♦❢ s❡❡ ❬✸✾(✳ ❚❤❡r❡❢♦r❡✱ ❜② ❋✉❜✐♥✐✬s t❤❡♦r❡♠✱ ϑ * T ❛❞♠✐ts t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ √ T (ϑ * T -ϑ) =⇒ 2J(ϑ) -1 R R F (ϑ, y) -✶ ■ {y>x} σ(y) f (ϑ, y) f (ϑ, x) ḟ (ϑ, x)❞x ❞W (y). ✭✸✳✶✸✮ ❍❡r❡ J (ϑ) = R ḟ (ϑ, x) 2 ❞x. ✭✸✳✶✹✮ ▼♦r❡♦✈❡r✱ ✉♥❞❡r r❡❣✉•❛r✐t② ❝♦♥❞✐t✐♦♥s R ❛♥❞ A 0 ✱ t❤❡ ▼❉❊ ϑ * T ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• ✭❢♦r ♠♦r❡ ❞❡t❛✐•s s❡❡ ❈❤❛♣t❡r ✷✱ ❙❡❝t✐♦♥ ✷✳✷ ✐♥ ❬✸✾(✮ L ϑ { √ T (ϑ * T -ϑ)} =⇒ L{ξ} = N (0, σ 2 (ϑ)), ✇❤❡r❡ σ 2 (ϑ) = J(ϑ) -2 R R f (ϑ, x) ḟ (ϑ, x) D ϑ (x, y) f (ϑ, y) ḟ (ϑ, y) ❞x ❞y. ❍❡r❡ D ϑ (x, y) = 4 E ϑ F (ϑ, ξ) -✶ ■ {ξ>x} F (ϑ, ξ) -✶ ■ {ξ>y} σ (ξ) 2 f (ϑ, ξ) 2 . ✸✳✹ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ st✉❞② t❤❡ ❆❉❋ •♦❋ t❡st ❜❛s❡❞ ♦♥ t❤❡ ❈r❛♠ér✲✈♦♥ ▼✐s❡s t②♣❡ st❛t✐st✐❝ δ T = ∞ -∞ η * T (ϑ * T , x) 2 ❞F (ϑ * T , x) , ✇❤❡r❡ η * T (ϑ * T , x) = √ T fT (x) -f (ϑ * T , x) . ✭✸✳✶✺✮ ✾✷ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❍❡r❡ ϑ * T ✐s t❤❡ ▼❉❊ ❞❡✜♥❡❞ ❜② ✭✸✳✶✶✮✳ ■♥tr♦❞✉❝❡ t❤❡ •❛✉ss✐❛♥ ❢✉♥❝t✐♦♥ η (ϑ, x) = 2 f (ϑ, x) +∞ -∞ F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) ❞W (y) -2J (ϑ) -1 R R F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) f (ϑ, x) ḟ (ϑ, x) ❞x ❞W (y) ḟ (ϑ, x). ✭✸✳✶✻✮ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✇❡ ❤❛✈❡ t❤✐s ❝♦♥✈❡r❣❡♥❝❡ ✭s❡❡ ❚❤❡♦r❡♠ ✸✳✹✳✺ ❜❡•♦✇✮ δ T = ∞ -∞ η * T (ϑ * T , x) 2 ❞F (ϑ * T , x) =⇒ δ ≡ ∞ -∞ η (ϑ, x) 2 ❞F (ϑ, x) . |❡♠❛r❦ t❤❛t t❤❡ •✐♠✐t st❛t✐st✐❝ δ ❞❡♣❡♥❞s str♦♥❣•② ♦♥ t❤❡ ♠♦❞❡•✱ ✐✳❡✳✱ f (•, •) ❛♥❞ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ✳ ❚❤❡r❡❢♦r❡ t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✐s ♥♦t ❆❉❋✳ ❚♦ ❛✈♦✐❞ t❤✐s ❞✐✣❝✉•t② ✇❡ ♣r♦♣♦s❡ ❛ ✜rst tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r η (•, •) ❣✐✈❡♥ ❛s ❢♦••♦✇s ✿ U (F (ϑ, x)) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) . ✭✸✳✶✼✮ |❡♠✐♥❞ t❤❛t t❤❡ ❛❜♦✈❡ tr❛♥s❢♦r♠❛t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✐♥ ❬✹✶(✳ ❋✉rt❤❡r✱ •❡t ✉s ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥s h (ϑ, s) = 2 J (ϑ) -1 C (ϑ) 1 2 Ṡ(ϑ, F -1 (ϑ, s)) σ (F -1 (ϑ, s)) ✭✸✳✶✽✮ ❛♥❞ g (ϑ, s) = C (ϑ) -1 2 1 0 s -✶■ {s>t} a (ϑ, s) b (ϑ, s) ḟ ϑ, F -1 (ϑ, t) ❞t, ✭✸✳✶✾✮ ✇❤❡r❡ a (ϑ, s) = σ (F -1 (ϑ, s))✱ b (ϑ, s) = f (ϑ, F -1 (ϑ, s)) , C (ϑ) = 1 0 1 0 v -✶■ {v>t} a (ϑ, v) b (ϑ, v) ḟ ϑ, F -1 (ϑ, t) ❞t 2 ❞v ✭✸✳✷✵✮ ❛♥❞ J (ϑ) = 1 0 ḟ (ϑ, F -1 (ϑ, s)) 2 f (ϑ, F -1 (ϑ, s))
❞s, x 2 φ+ 2 (ϑ * T , y) ψ2 (ϑ * T , y)

1 0 g (ϑ, s) 2 ❞s = 1. ❍❡r❡ F -1 (ϑ, s) ✐s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ F (ϑ, y)✱ ✐✳❡✳✱ y ✐s s♦•✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ s = F (ϑ, y)✳ ❚❤❡♥✱ ✇❡ s❤♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✸✳✶✼✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ U (t) = w (t) - 1 0 g (ϑ, s) ❞w (s) t 0 h (ϑ, s) ❞s, 0 ≤ t ≤ 1, ✭✸✳✷✶✮ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✾✸ ✇❤❡r❡ w (s) , 0 ≤ s ≤ 1 ✐s s♦♠❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ▼♦r❡♦✈❡r✱ ✇❡ s❤♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✸✳✶✼✮ ♦❢ η * T (•, •) ❞❡✜♥❡❞ ❜② ✭✸✳✶✺✮ ❣✐✈❡s ✉s ❛ st❛t✐st✐❝ ✇❤✐❝❤ ✐s ❛s②♠♣t♦t✐❝❛••② ❡q✉✐✈❛•❡♥t t♦ ξ T (ϑ * T , x) = 1 √ T T 0 ✶■ {Xs<x} σ (X s ) [❞X s -S (ϑ * T , X s ) ❞s] , ✭✸✳✷✷✮ ✐✳❡✳✱ ❜♦t❤ st❛t✐st✐❝s ❤❛✈❡ t❤❡ s❛♠❡ •✐♠✐t ♣r♦❝❡ss ✭✸✳✷✶✮✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤❡ st❛t✐st✐❝ ✭✸✳✷✷✮ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✸✹( ✇❤❡r❡ t❤❡ ▼▲❊ ✇❛s ✉s❡❞ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r✳ ❚❤❡r❡❢♦r❡✱ ♦✉r ❆❉❋ t❡st ✇✐•• ❜❡ ❜❛s❡❞ ♦♥ t❤❡ st❛t✐st✐❝ ξ T (•, •) ❢♦r s✐♠♣•✐❝✐t② ♦❢ ❝❛•❝✉•✉s✳ ❚❤❡ •❛st st❡♣ ✐s t♦ ❛♣♣•② t❤❡ s❡❝♦♥❞ tr❛♥s❢♦r♠❛t✐♦♥ L 2 [•] t♦ t❤❡ •✐♠✐t ♣r♦❝❡ss ✭✸✳✷✶✮ ✭❢r♦♠ ❚❤❡♦r❡♠ ✶✳✹✳✶✮ ❛♥❞ t♦ ♦❜t❛✐♥ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L 2 [U ] (t) = w t , 0 ≤ t ≤ 1. ❚❤❡♥ ✇❡ ❝❛♥ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡st ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥✳ ❇❡•♦✇ ✇❡ r❡❛•✐③❡ t❤✐s ♣r♦❣r❛♠✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦••♦✇✐♥❣ r❡s✉•t✳ ❚❤❡♦r❡♠ ✸✳✹✳✺✳ ▲❡t t❤❡ ❝♦♥❞✐t✐♦♥s R✱ ES ❛♥❞ A 0 ❜❡ ❢✉•✜••❡❞✱ t❤❡♥ η * T (ϑ * T , x) =⇒ η (ϑ, x) , ξ T (ϑ * T , x) =⇒ U (F (ϑ, x)) ✭✸✳✷✸✮ ❛♥❞ U (F (ϑ, x)) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) . Pr♦♦❢✳ ❚❤❡ st❛t✐st✐❝ η * T (•, •) ❞❡✜♥❡❞ ❜② ✭✸✳✶✺✮ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ η * T (ϑ * T , x) = √ T fT (x) -f (ϑ, x) - √ T (ϑ * T -ϑ) ḟ (ϑ, x) + o (1) . |❡♠✐♥❞ t❤❛t t❤❡ ♥♦r♠❛•✐③❡❞ ❞✐✛❡r❡♥❝❡ √ T ( fT (x) -f (ϑ, x)) ❤❛s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ✭✸✳✶✷✮ ❛♥❞ t❤❡ ▼❉❊ ϑ * T ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✳✶✸✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢♦••♦✇✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ ✭✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✮ η * T (ϑ * T , x) =⇒ η (ϑ, x) = 2 f (ϑ, x) +∞ -∞ F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) ❞W (y) -2J (ϑ) -1 R R F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) f (ϑ, x) ḟ (ϑ, x) ❞x ❞W (y) ḟ (ϑ, x) ✾✹ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❤♦•❞s✳ ❍❡r❡ ḟ (ϑ, x) = 2 f (ϑ, x) - Ġ (ϑ) 2 G (ϑ) + x 0 Ṡ (ϑ, y) σ (y) 2 ❞y . ❇❡•♦✇✱ ✇❡ s❤♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✸✳✶✼✮ ❛❞♠✐ts t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✳✷✹✮✳ ▲❡♠♠❛ ✸✳✹✳✻✳ ❲❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t② U (t) = w (t) - 1 0 g (ϑ, s) ❞w (s) t 0 h (ϑ, s) ❞s, ✭✸✳✷✹✮ ✇❤❡r❡ w (t) , 0 ≤ t ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ Pr♦♦❢✳ ❲❡ ❤❛✈❡ ❢♦••♦✇✐♥❣ ✭✸✳✶✼✮ ✿ U (F (ϑ, x)) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) = x -∞ σ (y) f (ϑ, y) ❞ ∞ -∞ F (ϑ, z) -✶ ■ {z>y} σ(z) f (ϑ, z) ❞W (z) -2J(ϑ) -1 R R F (ϑ, y) -✶ ■ {y>x} σ(y) f (ϑ, y) ḟ (ϑ, x) ❞F (ϑ, x) ❞W (y) x -∞ Ṡ(ϑ, y) σ(y) ❞F (ϑ, y). ❋✉rt❤❡r✱ x -∞ σ (y) f (ϑ, y) ❞ ∞ -∞ F (ϑ, z) -✶ ■ {z>y} σ(z) f (ϑ, z) ❞W (z) = x -∞ σ (y) f (ϑ, y) ❞ y -∞ F (ϑ, z) σ(z) f (ϑ, z) ❞W (z) + x -∞ σ (y) f (ϑ, y) ❞ ∞ y F (ϑ, z) -1 σ(z) f (ϑ, z) ❞W (z) = x -∞ f (ϑ, y) ❞W (y) . ❚❤❡r❡❢♦r❡✱ ✇❡ ♦❜t❛✐♥ U (F (ϑ, x)) = x -∞ f (ϑ, y) ❞W (y) -2J(ϑ) -1 R R F (ϑ, y) -✶ ■ {y>x} σ(y) f (ϑ, y) ḟ (ϑ, x) ❞F (ϑ, x) ❞W (y) x -∞ Ṡ(ϑ, y) σ(y) ❞F (ϑ, y). ▲❡t ✉s ❝❤❛♥❣❡ t❤❡ ✈❛r✐❛❜•❡s t = F (ϑ, x)✱ s = F (ϑ, y)✱ a(ϑ, s) = σ (F -1 (ϑ, s))✱ b(ϑ, s) = f (ϑ, F -1 (ϑ, s)) ❛♥❞ w(s) = F -1 (ϑ,s) -∞ f (ϑ, y) ❞W (y) , ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✾✺ ✇❤❡r❡ w (s) , 0 ≤ s ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❍❡♥❝❡✱ ✇❡ ❝❛♥ ✇r✐t❡ U (t) = w (t) -2 J(ϑ) -1 1 0 1 0 s -✶■ {s>t} a(ϑ, s)b(ϑ, s) ḟ (ϑ, F -1 (ϑ, t)) ❞t ❞w (s) t 0 Ṡ(ϑ, F -1 (ϑ, s)) σ(F -1 (ϑ, s)) ❞s = w (t) - 1 0 g (ϑ, s) ❞w (s) t 0 h (ϑ, s) ❞s. ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❝❡ss U (•) ❤❛s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✳✷✹✮ ❛♥❞ t❤❡ ▲❡♠♠❛ ✐s ♣r♦✈❡❞✳ ◆♦✇✱ ✇❡ s❤♦✇ t❤❛t t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✸✳✶✼✮ ♦❢ η * T (•, •) ❣✐✈❡s ✉s ❛ st❛t✐st✐❝ ✇❤✐❝❤ ✐s ❛s②♠♣t♦t✐❝❛••② ❡q✉✐✈❛•❡♥t t♦ ✭✸✳✷✷✮✳ ■♥❞❡❡❞✱ ✇❡ ❤❛✈❡ ξ T (ϑ * T , x) = 1 √ T T 0 ✶■ {Xs<x} σ (X s ) [❞X s -S (ϑ, X s ) ❞s] - 1 √ T T 0 ✶■ {Xs<x} σ (X s ) [S (ϑ * T , X s ) -S (ϑ, X s )] ❞s = 1 √ T T 0 ✶■ {Xs<x} ❞W s - √ T (ϑ * T -ϑ) 1 T T 0 ✶■ {Xs<x} Ṡ( θ, X s ) σ (X s ) ❞s + o(1). ❯s✐♥❣ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠✱ t❤❡ ✜rst st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ✐s ❛s②♠♣t♦t✐❝❛••② ♥♦r♠❛• 1 √ T T 0 ✶■ {Xs<x} ❞W s =⇒ x -∞ f (ϑ, y) ❞W (y) ∼ N (0, F (ϑ, x)) ❛♥❞ ❜② t❤❡ •❛✇ ♦❢ •❛r❣❡ ♥✉♠❜❡rs✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛• ✭❛s T → ∞✮ 1 T T 0 ✶■ {Xs<x} Ṡ (ϑ, X s ) σ (X s ) ❞s -→ x -∞ Ṡ (ϑ, y) σ (y) ❞F (ϑ, y) . ❚❤❡r❡❢♦r❡✱ ✉s✐♥❣ ✭✸✳✶✸✮ ❛♥❞ t❤❡ ❛❜♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉•ts✱ ♦♥❡ ❝❛♥ ✇r✐t❡ ξ T (ϑ * T , x) =⇒ x -∞ f (ϑ, y) ❞W (y) -2J(ϑ) -1 R R F (ϑ, y) -✶■ {y>x} σ(y) f (ϑ, y) ḟ (ϑ, x) ❞F (ϑ, x) ❞W (y) x -∞ Ṡ(ϑ, y) σ(y) ❞F (ϑ, y) = U (F (ϑ, x)) . ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ t❤❡ ❛s②♠♣t♦t✐❝ ❡q✉✐✈❛•❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✭✸✳✶✼✮ ♦❢ η * T (•, •) ❛♥❞ t❤❡ st❛t✐st✐❝ ✭✸✳✷✷✮✳ ❋✐♥❛••②✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✸✳✷✸✮✱ ✇❤✐❝❤ ♣r♦✈❡s t❤❡ t❤❡♦r❡♠✳ ✾✻ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❉✉❡ t♦ ❚❤❡♦r❡♠ ✸✳✹✳✺✱ ♦♥❡ ❝❛♥ ✇r✐t❡ ∆ = ∞ -∞ x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) 2 ❞F (ϑ, x) = 1 0 U (t) 2 ❞t. ■t ✐s ❡❛s✐•② s❡❡♥ t❤❛t ∆ T = ∞ -∞ x -∞ σ (y) f (ϑ * T , y) ❞ η * T (ϑ * T , y) 2f (ϑ * T , y) 2 ❞F (ϑ * T , x) ❛♥❞ t❤❡ st❛t✐st✐❝ ∆ * T = ∞ -∞ ξ T (ϑ * T , x) 2 ❞F (ϑ * T , x) ❤❛✈❡ t❤❡ s❛♠❡ •✐♠✐t ♣r♦❝❡ss ∆✳ ❚❤❡♥✱ ✐t ✐s ♦❜✈✐♦✉s t♦ s❡❡ t❤❛t t❤❡ t❡st ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ ❜❛s❡ ♦❢ t❤✐s st❛t✐st✐❝ ψ T = ✶ ■ {∆ * T > cα} , P (∆ > c α ) = α, α ∈ (0, 1) ✐s ♥♦t ❆❉❋✳ ❙♦ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L 2 [•] ❞❡✜♥❡❞ ❜② ✭✶✳✷✻✮ s✉❝❤ t❤❛t 1 0 L 2 [U ] (t) 2 ❞t = 1 0 w 2 t ❞t, 0 ≤ t ≤ 1. ❍❡r❡ t❤❡ ♣r♦❝❡ss U (•) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✷✹✮✳ ❋✐♥❛••②✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡st ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✉s✐♥❣ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥✑ ♦❢ t❤❡ t❡st st❛t✐st✐❝ ✇✐t❤ t❤❡ s❛♠❡ •✐♠✐t✳ ✸✳✺ ❚❡st ■♥ t❤✐s ❙❡❝t✐♦♥✱ ♦✉r ♣✉r♣♦s❡ ✐s t♦ t❡st t❤❡ ♣❛r❛♠❡tr✐❝ ❤②♣♦t❤❡s✐s H 0 ✳ ❚❤✉s✱ ✇❡ ✇✐•• ♣r♦♣♦s❡ ❛ st❛t✐st✐❝ ❜❛s❡❞ ♦♥ t❤❡ ▼❉❊ ϑ * T ✳ |❡❝❛•• t❤❛t t❤❡ st❛rt✐♥❣ st❛t✐st✐❝ η * T (ϑ * T , x) = √ T fT (x) -f (ϑ * T , x) ✭✸✳✷✺✮ ❝♦♥✈❡r❣❡s t♦ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ η (ϑ, x) ❞❡✜♥❡❞ ❜② ✭✸✳✶✻✮✳ ❚❤❡♥ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L 1 [η] (x) = x -∞ σ (y) f (ϑ, y) ❞ η (ϑ, y) 2f (ϑ, y) •❡❛❞s t♦ t❤❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ U (•) ❞❡✜♥❡❞ ❜② ✭✸✳✷✹✮ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ •✐♠✐t ♦❢ t❤❡ ♣r♦❝❡ss ✭✸✳✷✷✮✳ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✾✼ ❚♦ ✜♥✐s❤✱ ✇❡ ❛♣♣•② t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ L 2 [•] ❢r♦♠ ❚❤❡♦r❡♠ ✶✳✹✳✶ s✉❝❤ t❤❛t L 2 [U ] (t) = w t , 0 ≤ t ≤ 1. ◆♦✇✱ ✇❡ ❤❛✈❡ t♦ r❡❛•✐③❡ t❤❡ s✐♠✐•❛r tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ♣r♦❝❡ss ξ T (•, •) ❣✐✈❡♥ ❜② ✭✸✳✷✷✮ ❛♥❞ t❤❡♥ ✇❡ s❤♦✇ t❤❛t t❤✐s st❛t✐st✐❝ ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss L 2 [ξ T ] (x) =⇒ w t , 0 ≤ t ≤ 1. ❚❤❡r❡❢♦r❡ t❤❡ t❡st ψ * T = ✶■ {δ * T >cα} ✇✐t❤ δ * T = ∞ -∞ L 2 [ξ T ] (x) 2 dF (ϑ * T , x) =⇒ 1 0 w 2 t dt ✇✐•• ❜❡ ❆❉❋ ❜❡❝❛✉s❡ t❤❡ •✐♠✐t ❞✐str✐❜✉t✐♦♥ ♦❢ δ * T ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ S (•, •)✱ σ (•) ❛♥❞ ϑ✳ ▲❡t ✉s r❡❛•✐③❡ t❤✐s ♣r♦❣r❛♠✳ ❉❡♥♦t❡ t❤❡ ❢✉♥❝t✐♦♥s ĥ (ϑ, z) = 2 J (ϑ) -1 C (ϑ) 1 2 Ṡ (ϑ, z) σ (z) ✭✸✳✷✻✮ ❛♥❞ ĝ (ϑ, z) = C (ϑ) -1 2 R F (ϑ, z) -✶■ {z>y} σ(z)f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y. ✭✸✳✷✼✮ ❍❡r❡ J (ϑ) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✶✹✮ ❛♥❞ C (ϑ) = R R F (ϑ, z) -✶■ {z>y} σ(z)f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y 2 f (ϑ, z) ❞z. ✭✸✳✷✽✮ ■♥tr♦❞✉❝❡ Î1 = y -∞ ĝ (ϑ, z) 2 f (ϑ, z) ❞z, Î2 = y -∞ ĥ (ϑ, z) ĝ (ϑ, z) f (ϑ, z) ❞z, Î3 = y -∞ ĥ (ϑ, z) f (ϑ, z) ❞z, Î4 = y -∞ ĥ (ϑ, z) 2 f (ϑ, z) ❞z ❛♥❞ ♣✉t Î5 = y -∞ ĝ (ϑ, z) f (ϑ, z) ❞z. ✾✽ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ■♥❞❡❡❞✱ ✇❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥s φ1 (•, •)✱ φ2 (•, •) ❛♥❞ ψ2 (•, •) ❣✐✈❡♥ ❜② ✭✶✳✷✸✮✱ ✭✶✳✷✹✮ ❛♥❞ ✭✶✳✷✺✮✱ r❡s♣❡❝t✐✈❡•②✱ ✇❤❡r❡ ✇❡ r❡♣•❛❝❡ g (•, •) ❛♥❞ h (•, •) ❜② ĝ = ĝ (ϑ, y) ❛♥❞ ĥ = ĥ (ϑ, y)✳ ❚❤❡♥ ✇❡ ✐♥tr♦❞✉❝❡ w F (ϑ,x) = U (F (ϑ, x)) ✭✸✳✷✾✮ + x -∞ y -∞ φ2 (ϑ, y) -1 [ φ1 (ϑ, y) ĥ(ϑ, z) + ψ2 (ϑ, y)ĝ(ϑ, z)] ❞U (F (ϑ, z)) f (ϑ, y) ❞y. ❚♦ ❝♦♥str✉❝t t❤❡ t❡st✱ ✇❡ ❤❛✈❡ t♦ r❡♣•❛❝❡ U (F (ϑ, x)) ❜② ξ T (ϑ * T , x) ❞❡✜♥❡❞ ❜② ✭✸✳✷✷✮ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s ĥ (•, •)✱ ĝ (•, •)✱ φ1 (•, •)✱ φ2 (•, •) ❛♥❞ ψ2 (•, •) ❜② t❤❡✐r ✏❡♠✲ ♣✐r✐❝❛• ✈❡rs✐♦♥s✑ ❜❛s❡❞ ♦♥•② ♦♥ t❤❡ ♦❜s❡r✈❛t✐♦♥s✳ ■♥ ❢❛❝t✱ ✇❡ ✐♥s❡rt t❤❡ ❡st✐♠❛t♦r ϑ * T ❣✐✈❡♥ ❜② ✭✸✳✶✶✮ ✐♥st❡❛❞ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡r ϑ ✐♥ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ t❤❡ ❛❜♦✈❡ ❢✉♥❝t✐♦♥s✳ ■♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ t❡st✱ ✇❡ ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ ❛♥ ❛❞❞✐t✐♦♥❛• ❝♦♥❞✐t✐♦♥✳ R 5 . ❚❤❡ ❢✉♥❝t✐♦♥ S (ϑ, x) ❤❛s t✇♦ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡s ✇✳r✳t✳ ϑ s✉❝❤ t❤❛t Ṡ (ϑ, x) , S (ϑ, x) ∈ P ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s Ṡ (ϑ, x) ❛♥❞ σ (x) ❤❛✈❡ ❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐✈❡s ✇✳r✳t✳ x s✉❝❤ t❤❛t Ṡ′ (ϑ, x) , σ ′ (x) ∈ P. ▼♦r❡♦✈❡r✱ ❞✉❡ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ▼❉❊✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭❛s T → ∞) φ2 (ϑ * T , y) -φ2 (ϑ, y) -→ 0. ❍❡♥❝❡✱ ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥ φ+ 2 (ϑ * T , y) =    φ2 (ϑ * T , y) -1 , ✐❢ φ2 (ϑ * T , y) > 0, 0, ❡•s❡, ✇❤✐❝❤ ❛s②♠♣t♦t✐❝❛••② ❝♦✐♥❝✐❞❡s ✇✐t❤ φ2 (ϑ, y) -1 ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ •✐♠✐t ❞✐str✐❜✉✲ t✐♦♥ ❞♦❡s ♥♦t ❝❤❛♥❣❡❞✳ ❲❡ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡st ❜❛s❡❞ ♦♥ t❤❡ ❢♦••♦✇✐♥❣ st❛t✐st✐❝ ✿ W * T (x) = ξ T (ϑ * T , x) + x -∞ y -∞ φ+ 2 (ϑ * T , y) φ1 (ϑ * T , y) ĥ (ϑ * T , z) + ψ2 (ϑ * T , y) ĝ (ϑ * T , z) ❞ξ T (ϑ * T , z) ❞F (ϑ * T , y) . ✭✸✳✸✵✮ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✾✾ ❚❤❡♥ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t W * T (x) =⇒ L 2 [U ] (t) = w t . ❚❤❡ ♠❛✐♥ t❡❝❤♥✐❝❛• ♣r♦❜•❡♠ ✐♥ t❤❡ r❡❛•✐③❛t✐♦♥ ♦❢ t❤✐s ♣r♦❣r❛♠ ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❢♦••♦✇✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ✿ N (ϑ * T , y) = y -∞ ĥ (ϑ * T , z) ❞ξ T (ϑ * T , z) ✭✸✳✸✶✮ ❛♥❞ M (ϑ * T , y) = y -∞ ĝ (ϑ * T , z) ❞ξ T (ϑ * T , z) . ✭✸✳✸✷✮ |❡♠❛r❦ t❤❛t ✇❡ ❝❛♥♥♦t ❝♦♠♣✉t❡ t❤❡s❡ ✐♥t❡❣r❛•s ❞✐r❡❝t•② ❜❡❝❛✉s❡ t❤❡ ✐♥t❡❣r❛♥❞s ❝♦♥t❛✐♥ t❤❡ ▼❉❊ ϑ * T ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ✇❤♦•❡ tr❛•❡❝t♦r② X T = {X t , 0 ≤ t ≤ T }✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ✭✸✳✸✶✮ ❛♥❞ ✭✸✳✸✷✮ ❛r❡ ♥♦t ✇❡•• ❞❡✜♥❡❞✳ ❚♦ ❛✈♦✐❞ t❤✐s ♣r♦❜•❡♠ ✇❡ r❡♣•❛❝❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s ❜② t❤❡ ♦r❞✐♥❛r② ♦♥❡s ✉s✐♥❣ t❤❡ ■tô ❢♦r♠✉•❛✳ ❲❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ s✐♠✐•❛r ♣r♦❜•❡♠ ✐♥ ❬✸✹( ❛♥❞ ❬✹✹(✳ ▲❡t ✉s s✉♣♣♦s❡ t❤❛t ĥ (•, •) ❛♥❞ ĝ (•, •) ❛r❡ ♣✐❡❝❡✲✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❝❛•❝✉•❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛•s b a ĥ (ϑ, z) ❞ξ T (ϑ, z) , b a ĝ (ϑ, z) ❞ξ T (ϑ, z) . ❋♦r ❛♥② ♣❛rt✐t✐♦♥ a = z 1 < z 2 < • • • < z m = b✱ ✇❡ ❝❛♥ ✇r✐t❡ m-1 k=1 ĥ (ϑ, zk ) [ξ T (ϑ, z k+1 ) -ξ T (ϑ, z k )] = 1 √ T T 0 m-1 k=1 ĥ (ϑ, zk ) σ (X s ) ✶■ {z k ≤Xs<z k+1 } ❞X s - 1 √ T T 0 m-1 k=1 ĥ (ϑ, zk ) σ (X s ) ✶■ {z k ≤Xs<z k+1 } S (ϑ, X s ) ❞s. ❚❤❡r❡❢♦r❡ ❛s max |z k+1 -z k | → 0✱ ✇❡ ♦❜t❛✐♥ t❤❡ •✐♠✐t lim m→∞ m-1 k=1 ĥ (ϑ, zk ) [ξ T (ϑ, z k+1 ) -ξ T (ϑ, z k )] = 1 √ T T 0 ĥ (ϑ, X s ) σ (X s ) ✶■ {a≤Xs<b} ❞X s - 1 √ T T 0 ĥ (ϑ, X s ) σ (X s ) ✶■ {a≤Xs<b} S (ϑ, X s ) ❞s ✶✵✵ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❛♥❞ ❢♦r a = -∞ ❛♥❞ b = y ✭♦✉r ❝❛s❡✮✱ ✇❡ ❤❛✈❡ t❤❡ ❡q✉❛•✐t② y -∞ ĥ (ϑ, z) ❞ξ T (ϑ, z) = 1 √ T T 0 ĥ (ϑ, X s ) σ (X s ) ✶■ {Xs<y} ❞X s - 1 √ T T 0 ĥ (ϑ, X s ) σ (X s ) ✶■ {Xs<y} S (ϑ, X s ) ❞s. ❇② ✉s✐♥❣ t❤❡ s❛♠❡ ❛r❣✉♠❡♥t✱ ✇❡ ♦❜t❛✐♥ y -∞ ĝ (ϑ, z) ❞ξ T (ϑ, z) = 1 √ T T 0 ĝ (ϑ, X s ) σ (X s ) ✶■ {Xs<y} ❞X s - 1 √ T T 0 ĝ (ϑ, X s ) σ (X s ) ✶■ {Xs<y} S (ϑ, X s ) ❞s. ✭✸✳✸✸✮ ◆♦✇✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢✉♥❝t✐♦♥ F (ϑ, y, x) = x X 0 R (ϑ, y, z) ❞z, ✇❤❡r❡ R (ϑ, y, z) = 2 J (ϑ) -1 C (ϑ) 1 2 Ṡ (ϑ, z) σ (z) 2 ✶■ {z<y} . ❍❡r❡ C (ϑ) ❛♥❞ J (ϑ) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✸✳✷✽✮ ❛♥❞ ✭✸✳✶✹✮✱ r❡s♣❡❝t✐✈❡•②✳ ❇② t❤❡ ■tô ❢♦r♠✉•❛✱ ❞F (ϑ, y, X s ) = F ′ x (ϑ, y, X s ) ❞X s - 1 2 F ′′ xx (ϑ, y, X s ) σ (X s ) 2 ❞s, ✇❡ ❝❛♥ ✇r✐t❡ T 0 R (ϑ, y, X s ) ❞X s = X T X 0 R (ϑ, y, z) ❞z - 1 2 T 0 R ′ x (ϑ, y, X s ) σ (X s ) 2 ❞s. ■♥❞❡❡❞✱ ✇❡ ♦❜t❛✐♥ y -∞ ĥ (ϑ, z) ❞ξ T (ϑ, z) = 1 √ T X T X 0 R (ϑ, y, z) ❞z - 1 √ T T 0 R (ϑ, y, X s ) S (ϑ, X s ) + 1 2 R ′ x (ϑ, y, X s ) σ (X s ) 2 ❞s. ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ ♥♦ ♠♦r❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛• ❛♥❞ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ t❤❡ ▼❉❊ ϑ * T ❣✐✈❡♥ ❜② ✭✸✳✶✶✮✳ ◆♦✇✱ t❤❡ ♣r♦❝❡ss ✭✸✳✸✶✮ ✐s ✇❡•• ❞❡✜♥❡❞ ❛♥❞ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ ❡①♣r❡ss✐♦♥ N (ϑ * T , y) = 1 √ T X T X 0 R (ϑ * T , y, z) ❞z - 1 √ T T 0 R (ϑ * T , y, X s ) S (ϑ * T , X s ) + 1 2 R ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s. ✭✸✳✸✹✮ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✶✵✶ ❯s✐♥❣ ♦♥❝❡ ♠♦r❡ t❤❡ ■tô ❢♦r♠✉•❛✱ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✳✸✸✮ ❜❡❝♦♠❡s ❛s ❢♦••♦✇s ✿ y -∞ ĝ (ϑ, z) ❞ξ T (ϑ, z) = 1 √ T X T X 0 Q (ϑ, y, z) ❞z - 1 √ T T 0 Q (ϑ, y, X s ) S (ϑ, X s ) + 1 2 Q ′ x (ϑ, y, X s ) σ (X s ) 2 ❞s. ❍❡r❡ Q (ϑ, y, z) = C (ϑ) -1 2 σ(z) R F (ϑ, z) -✶■ {z>y} σ(z)f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y ✶■ {z<y} , ✇❤❡r❡ C (ϑ) ✐s ❣✐✈❡♥ ❜② ✭✸✳✷✽✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❝❡ss ✭✸✳✸✷✮ ✐s ✇❡•• ❞❡✜♥❡❞ ❛♥❞ ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ M (ϑ * T , y) = 1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 Q (ϑ * T , y, X s ) S (ϑ * T , X s ) + 1 2 Q ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s. ✭✸✳✸✺✮ ◆♦✇✱ ✉s✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ✭✸✳✸✹✮ ❛♥❞ ✭✸✳✸✺✮✱ t❤❡ st❛t✐st✐❝ ✭✸✳✸✵✮ ❜❡❝♦♠❡s ❛s ❢♦••♦✇s ✿ W * * T (x) = ξ T (ϑ * T , x) + x -∞ φ+ 2 (ϑ * T , y) φ1 (ϑ * T , y) 1 √ T X T X 0 R (ϑ * T , y, z) ❞z - 1 √ T T 0 R(ϑ * T , y, X s )S(ϑ * T , X s ) + 1 2 R ′ x (ϑ * T , y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y) + x -∞ φ+ 2 (ϑ * T , y) ψ2 (ϑ * T , y) 1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 Q(ϑ * T , y, X s )S(ϑ * T , X s ) + 1 2 Q ′ x (ϑ * T , y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y). ✭✸✳✸✻✮ ▲❡t ✉s ♣✉t δ * T = ∞ -∞ W * * T (x) 2 ❞F (ϑ * T , x) . ❚❤❡ ♠❛✐♥ r❡s✉•t ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ❣✐✈❡♥ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❚❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✸✳✺✳✻✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ES✱ A 0 ✱ R✱ R 4 ❛♥❞ R 5 ❛r❡ ❢✉•✜••❡❞✱ t❤❡♥ t❤❡ t❡st ψ * T = ✶■ {δ * T >cα} , P 1 0 w 2 t ❞t > c α = α ✐s ❆❉❋ ❛♥❞ ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1)✳ ✶✵✷ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ Pr♦♦❢✳ ■t ✐s ❡♥♦✉❣❤ t♦ ✈❡r✐❢② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ δ * T =⇒ ∞ -∞ w 2 F (ϑ,x) ❞F (ϑ * T , x) ≡ δ * ✭✸✳✸✼✮ ✉♥❞❡r ❤②♣♦t❤❡s✐s H 0 ✳ ❍❡r❡ w F (ϑ,x) ✐s ❣✐✈❡♥ ❜② ✭✸✳✷✾✮✳ ◆♦t❡ t❤❛t ❞✉❡ t♦ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ▼❉❊ ϑ * T ✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ (❛s T → ∞) ĥ (ϑ * T , z) -→ ĥ (ϑ, z) ĝ (ϑ * T , z) -→ ĝ (ϑ, z) , J (ϑ * T ) -→ J (ϑ) , ψ2 (ϑ * T , z) -→ ψ2 (ϑ, z) ❛♥❞ φ1 (ϑ * T , z) -→ φ1 (ϑ, z), φ2 (ϑ * T , z) -→ φ2 (ϑ, z). |❡♠✐♥❞ t❤❛t ✇❡ ♣r♦✈❡❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✐♥ ❚❤❡♦r❡♠ ✸✳✹✳✺✮ ξ T (ϑ * T , x) =⇒ U (F (ϑ, x)) . ❚❤❡♥ ✇❡ ❤❛✈❡ t♦ s❤♦✇ t❤❛t N (ϑ * T , y) =⇒ N (ϑ, y) , ✭✸✳✸✽✮ ✇❤❡r❡ N (ϑ * T , y) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✸✹✮ ❛♥❞ N (ϑ, y) = y -∞ ĥ (ϑ, z) ❞U (F (ϑ, z)) = J(ϑ) -1 C(ϑ) 1 2 y -∞ Ṡ(ϑ, z) σ(z) f (ϑ, z)❞W (z) ✭✸✳✸✾✮ - 4 C(ϑ) 1 2 J(ϑ) 2 R R F (ϑ, z) -✶■ {z>y} σ(z) f (ϑ, z) f (ϑ, y) ḟ (ϑ, y)❞y❞W (z) y -∞ Ṡ(ϑ, z) 2 σ(z) 2 f (ϑ, z)❞z. ❍❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ĥ (ϑ, z) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✷✻✮✳ ❚❤❡ ♣r♦❝❡ss N (ϑ * T , y) ❤❛s t❤❡ ❢♦••♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ N (ϑ * T , y) = 1 √ T X T X 0 R (ϑ * T , y, z) ❞z - 1 √ T T 0 R (ϑ * T , y, X s ) S (ϑ * T , X s ) + 1 2 R ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s = 1 √ T X T X 0 R (ϑ * T , y, z) ❞z ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✶✵✸ - 1 √ T T 0 R (ϑ * T , y, X s ) S (ϑ, X s ) + 1 2 R ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s - 1 √ T T 0 R (ϑ * T , y, X s ) [S (ϑ * T , X s ) -S (ϑ, X s )] ❞s = I * T (ϑ * T , y) -K * T (ϑ * T , y) . ❈♦♠❜✐♥✐♥❣ ■tô✬s ❢♦r♠✉•❛ ❛♥❞ t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s✱ ✇❡ ♦❜t❛✐♥ I * T (ϑ * T , y) = 1 √ T X T X 0 R (ϑ * T , y, z) ❞z - 1 √ T T 0 R (ϑ * T , y, X s ) S (ϑ, X s ) + 1 2 R ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s = 1 √ T T 0 R (ϑ, y, X s ) σ (X s ) ❞W s + o(1) =⇒ ∞ -∞ R (ϑ, y, z) σ (z) f (ϑ, z) ❞W (z) , ✇❤❡r❡ W (•) ✐s ❞♦✉❜•❡✲s✐❞❡❞ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❋✉rt❤❡r✱ ❜② ❚❛②•♦r✬s ❢♦r♠✉•❛✱ t❤❡ •❛✇ ♦❢ •❛r❣❡ ♥✉♠❜❡rs ❛♥❞ ✉s✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✳✶✸✮✱ ✇❡ ❝❛♥ ✇r✐t❡ K * T (ϑ * T , y) = 1 √ T T 0 R (ϑ * T , y, X s ) [S (ϑ * T , X s ) -S (ϑ, X s )] ❞s = √ T (ϑ * T -ϑ) 1 T T 0 R (ϑ * T , y, X s ) Ṡ( θ, X s ) ❞s =⇒ 2 J (ϑ) -1 R R F (ϑ, z) -✶■ {z>y} σ(z) f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y ❞W (z) ∞ -∞ R (ϑ, y, z) Ṡ(ϑ, z) f (ϑ, z) ❞z. ❚❤❡r❡❢♦r❡✱ ✇❡ ♦❜t❛✐♥ N (ϑ * T , y) =⇒ N (ϑ, y) , ✇❤❡r❡ N (ϑ, y) ✐s ❣✐✈❡♥ ❜② ✭✸✳✸✾✮✳ ■♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✱ ✇❡ s❤♦✇ t❤❛t ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ M (ϑ * T , y) =⇒ M (ϑ, y) , ✭✸✳✹✵✮ ✇❤❡r❡ M (ϑ * T , y) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✸✺✮ ❛♥❞ M (ϑ, y) = y -∞ ĝ (ϑ, z) ❞U (F (ϑ, z)) = C (ϑ) -1 2 y -∞ R F (ϑ, z) -✶■ {z>y} σ(z)f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y f (ϑ, z) ❞W (z) -4 J (ϑ) -1 C(ϑ) -1 2 R R F (ϑ, z) -✶■ {z>y} σ(z) f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y ❞W (z) y -∞ R F (ϑ, z) -✶■ {z>y} σ(z) 2 f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y Ṡ(ϑ, z) f (ϑ, z) ❞z. ✭✸✳✹✶✮ ✶✵✹ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❍❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ĝ (ϑ, z) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✷✼✮✳ ❋♦r t❤❡ ♣r♦❝❡ss M (ϑ * T , y)✱ ✇❡ ❤❛✈❡ M (ϑ * T , y) = 1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 Q (ϑ * T , y, X s ) S (ϑ * T , X s ) + 1 2 Q ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s = 1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 Q (ϑ * T , y, X s ) S (ϑ, X s ) + 1 2 Q ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s - 1 √ T T 0 Q (ϑ * T , y, X s ) [S (ϑ * T , X s ) -S (ϑ, X s )] ❞s = Ĩ * T (ϑ * T , y) -K * T (ϑ * T , y) . ❋♦r t❤❡ ✜rst ✐♥t❡❣r❛•✱ ❞✉❡ t♦ t❤❡ ■tô ❢♦r♠✉•❛ ❛♥❞ t❤❡ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠ ❢♦r st♦❝❤❛st✐❝ ✐♥t❡❣r❛•s✱ ✇❡ ❝❛♥ ✇r✐t❡ Ĩ * T (ϑ * T , y) = 1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 Q (ϑ * T , y, X s ) S (ϑ, X s ) + 1 2 Q ′ x (ϑ * T , y, X s ) σ (X s ) 2 ❞s = 1 √ T T 0 Q (ϑ, y, X s ) σ (X s ) ❞W s + o(1) =⇒ ∞ -∞ Q (ϑ, y, z) σ (z) f (ϑ, z) ❞W (z) . ❋✉rt❤❡r✱ ✇❡ ❤❛✈❡ ❢♦r t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛• K * T (ϑ * T , y) = 1 √ T T 0 Q (ϑ * T , y, X s ) [S (ϑ * T , X s ) -S (ϑ, X s )] ❞s = √ T (ϑ * T -ϑ) 1 T T 0 Q (ϑ * T , y, X s ) Ṡ( θ, X s ) ❞s =⇒ 2 J (ϑ) -1 R R F (ϑ, z) -✶■ {z>y} σ(z) f (ϑ, z) f (ϑ, y) ḟ (ϑ, y) ❞y ❞W (z) ∞ -∞ Q (ϑ, y, z) Ṡ(ϑ, z) f (ϑ, z) ❞z. ❚❤❡r❡❢♦r❡ M (ϑ * T , y) =⇒ M (ϑ, y)✱ ✇❤❡r❡ M (ϑ, y) ✐s ❞❡✜♥❡❞ ❜② ✭✸✳✹✶✮✳ ❋✐♥❛••②✱ ✭✸✳✷✸✮✱ ✭✸✳✸✽✮ ❛♥❞ ✭✸✳✹✵✮ ❣✐✈❡ ✉s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ W * * T (x) =⇒ w F (ϑ,x) . ◆♦✇✱ t♦ ♦❜t❛✐♥ ✭✸✳✸✼✮✱ ✇❡ ✇✐•• ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♥t❡❣r❛•s ❜② ❝❤❡✲ ❝❦✐♥❣ t❤❡ ❢♦••♦✇✐♥❣ t❤r❡❡ ❝♦♥❞✐t✐♦♥s ✿ ✭s❡❡ ❬✷✼( ❛♥❞ ❬✸✹(✮ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ✶✵✺ ✶✳ ❋♦r ❛♥② x 1 , . . . , x k (W * * T (x 1 ) , . . . , W * * T (x k )) =⇒ w F (ϑ,x 1 ) , . . . , w F (ϑ,x k ) , ✇❤❡r❡ W * * T (x) ❛♥❞ w F (ϑ,x) ❛r❡ ❞❡✜♥❡❞ ❜② ✭✸✳✸✻✮ ❛♥❞ ✭✸✳✷✾✮✱ r❡s♣❡❝t✐✈❡•②✳ ✷✳ ❋♦r |x i | < L, i = 1, 2 E ϑ |W * * T (x 1 ) -W * * T (x 2 )| 2 ≤ C |x 1 -x 2 | 1 2 , ✭✸✳✹✷✮ ✇❤❡r❡ C ❛♥❞ L ❛r❡ s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✳ ✸✳ ❋♦r ❛♥② κ > 0 t❤❡r❡ ❡①✐st L > 0 s✉❝❤ t❤❛t |x|>L E ϑ W * * T (x) 2 f (ϑ * T , x) ❞x < κ. ✭✸✳✹✸✮ ❚❤❡ ✜rst ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛• ❞✐str✐❜✉t✐♦♥s ❢♦••♦✇s ❢r♦♠ ✭✸✳✷✸✮✱ ✭✸✳✸✽✮ ❛♥❞ ✭✸✳✹✵✮ ❢♦r ❛♥② x 1 , . . . , x k ✳ ❚❤❡♥✱ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❡st✐♠❛t❡ ✭✸✳✹✷✮✱ ✇❡ ✇r✐t❡
1 √ T X T X 0 Q (ϑ * T , y, z) ❞z - 1 √ T T 0 [Q(ϑ * T , y, X s )S(ϑ * T , X s ) + 1 2 Q ′ x (ϑ * T , y, X s )σ(X s ) 2 ]❞s ❞F (ϑ * T , y) 2 .
❋✉rt❤❡r✱ ❜② ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t②✱ ✇❡ ❤❛✈❡ x 2 f (ϑ, y) ❞y + C

E ϑ |ξ T (ϑ * T , x 1 ) -ξ T (ϑ * T , x 2 )| 2 ≤ 2 E ϑ 1 √ T T 0 ✶■ {x 2 <Xs<x 1 } ❞W s 2 + 2 E ϑ √ T (ϑ * T -ϑ) 1 T T 0 Ṡ( θ, X s ) σ (X s ) ✶■ {x 2 <Xs<x 1 } ❞s 2 ≤ 2 x 1 x 2 f (ϑ, y) ❞y + 2 E ϑ √ T (ϑ * T -ϑ)
x 1

x 2 P (y) f (ϑ, y) ❞y

Mots Clés

estimateur de distance minimale, tests asymptotically distribution free, processus stochastiques, processus de diffusion, processus de Poisson non homogène.

Abstract

This work is devoted to the problem of the construction of several goodness of-fit (GoF) tests in the case of some stochastic processes observed in continuous time. As models of observations, we take processes and an inhomogeneous Poisson process. Under the null hypothesis, we treat the case where each model depends on an unknown one-dimensional parameter and we consider the minimum distance estimator for this parameter. Our goal is a (ADF) GoF tests of asymptotic size in the case of the parametric null hypotheses for the considered models. Indeed, we show that the limit of each studied statistic does not depend on the model and the unknown parameter. Therefore, the tests based on these statistics are ADF. The main purpose of this work is to construct a special linear transformation. In particular, we solve Fredholm equation of the second kind with degenerated kernel. Its solution gives us the desired linear transformation. Next, we show that the application of this transformation to the basic statistics allows us to introduce statistics with the same limit (the integral of the square of the because it does not depend on the models and the unknown parameter. Therefore, we construct the ADF GoF tests which are based on this l and ergodic) and inhomogeneous Poisson processes.

Key Words

Goodness-of-fit tests, minimum distance estimator, asymptotically distribution free tests, stochastic processes, diffusion process, and inhomogeneous Poisson process.

Bretagne Loire

ajustement pour des processus stochastiques dans le cas

Maroua BEN ABDEDDAIEM

  ❛ r❡♣rés❡♥t❛t✐♦♥ s✉✐✈❛♥t❡ ✿ ■◆❚|❖❉❯❈❚■❖◆ ✼ ❈❛s ❞❡s ♦❜s❡r✈❛t✐♦♥s ✐✳✐✳❞✳ ❛✈❡❝ ✉♥ ❡st✐♠❛t❡✉r ❞✐❢✲ ❢ér❡♥t ❞❡ •✬❊▼❱ ❙♦✉•✐❣♥♦♥s q✉❡ ❞❛♥s ❞❡ ♥♦♠❜r❡✉① tr❛✈❛✉①✱ •✬❡st✐♠❛t❡✉r ✉t✐•✐sé ♣♦✉r •❡ ♣❛r❛✲ ♠ètr❡ ✐♥❝♦♥♥✉ ét❛✐t t♦✉•♦✉rs •✬❊▼❱ ❡t ❝❡❝✐ ❡st ✐♠♣♦rt❛♥t ♣♦✉r •❛ ❝♦♥str✉❝t✐♦♥ ❞❡ •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❡t ❞✉ t❡st ❆❉❋✳ ❈❡♣❡♥❞❛♥t✱ ❞❛♥s ❝❡rt❛✐♥s ♣r♦❜•è♠❡s✱ ✐• ❡st ✐♠♣♦ss✐❜•❡ ❞✬❛✈♦✐r ✉♥❡ ❡①♣r❡ss✐♦♥ ❡①♣•✐❝✐t❡ ♣♦✉r •✬❊▼❱ ❡t ❞♦♥❝ ✐• ❡st ♣❛r❢♦✐s ♣ré❢ér❛❜•❡ ❞✬✉t✐•✐s❡r ❞✬❛✉tr❡s ❡st✐♠❛t❡✉rs q✉✐ ♣❡✉✈❡♥t êtr❡ ❢❛❝✐•❡♠❡♥t ❝❛•❝✉•és✳ P❛r ❡①❡♠♣•❡✱ ✐• ♣♦✉rr❛✐t êtr❡ •✬❡st✐♠❛t❡✉r ❞❡ ❞✐st❛♥❝❡ ♠✐♥✐♠❛•❡ ✭❊❉▼✮✱ •✬❡st✐♠❛t❡✉r ❞❡ ♠ét❤♦❞❡ ❞❡s ♠♦♠❡♥ts ♦✉ •✬❡st✐♠❛t❡✉r ❞❡ ✏tr❛•❡❝t♦r② ✜tt✐♥❣✑ ❡t❝✳ ❉❛♥s t♦✉s ❝❡s ❝❛s✱ •✬❡①♣r❡ss✐♦♥ ❞❡s st❛t✐st✐q✉❡s •✐♠✐t❡s s❡r❛ ❝♦♠♠❡ ✭✸✮ ✭❝❛s ❞❡ •✬❊▼❱✮ ♠❛✐s ❛✈❡❝ ❞❡✉① ❢♦♥❝t✐♦♥s ❞✐✛ér❡♥t❡s h (•, •) ❡t g (•, •) ✭✈♦✐r ✭✻✮ ❝✐✲❞❡ss♦✉s✮✳

  ϑ, s) ❞s, ❛✈❡❝ h 1 (ϑ, s) = g 1 (ϑ, s) = 1✱ h 2 (ϑ, s) = h 3 (ϑ, s) = h (ϑ, s)✱ g 2 (ϑ, s) = g (ϑ, s) ❡t g 3 (ϑ, s) = -1 0 g (ϑ, s) ❞s✱ ❝❡ q✉✐ ✈❛ ❛✉❣♠❡♥t❡r •❛ ❞✐♠❡♥s✐♦♥ ❞❡s ❢♦♥❝t✐♦♥s ✶✵ ■◆❚|❖❉❯❈❚■❖◆

  r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✳ ◆♦t♦♥s q✉❡ ❝❡tt❡ r❡♣rés❡♥t❛t✐♦♥ ♣❡✉t êtr❡ ♦❜t❡♥✉❡ ♣♦✉r ✉♥❡ •❛r❣❡ ❝•❛ss❡ ❞✬❡st✐♠❛t❡✉rs✳ ▲❛ ❞✐✛ér❡♥❝❡ ♣r✐♥❝✐♣❛•❡ ❛✈❡❝ •❡ ❝❛s ✐✳✐✳❞✳ ❡st ❞û ❛✉ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r ❡t ❞♦♥❝ •❛ ♣♦ss✐❜✐•✐té ❞❡ ❝♦♥str✉✐r❡ ✉♥❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❛♣♣•✐q✉é❡ ❛✉ ♣r♦❝❡ss✉s •✐♠✐t❡ ✭✼✮✳ P❛r ❝♦♥séq✉❡♥t✱ ♥♦✉s ❛••♦♥s ✈♦✐r ❞❛♥s ❝❡ tr❛✈❛✐• q✉❡ ♥♦✉s ♦❜t❡♥♦♥s •❛ ♠ê♠❡ •✐♠✐t❡ ✭✼✮ ❡♥ ✉t✐•✐s❛♥t •✬❊❉▼ ❞❛♥s •❡ ❝❛s ❞❡s ♣r♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑ ❡t ❡r❣♦❞✐q✉❡ ❡t ❞❛♥s •❡ ❝❛s ❞❡ ♣r♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡✳ |és✉•t❛ts ♣r✐♥❝✐♣❛✉① ▲❡ ❜✉t ♣r✐♥❝✐♣❛• ❞❡ ❝❡ tr❛✈❛✐• ❡st •❛ ❝♦♥str✉❝t✐♦♥ ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ♣♦✉r •❡s ♣r♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ✭❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑ ❡t ❡r❣♦❞✐q✉❡✮ ❡t •❡ ♣r♦❝❡ss✉s ❞❡ P♦✐s✲ s♦♥ ♥♦♥ ❤♦♠♦❣è♥❡ ♦❜s❡r✈és ❡♥ t❡♠♣s ❝♦♥t✐♥✉✳ ❙♦✉s •✬❤②♣♦t❤ès❡ ♥✉••❡✱ ♥♦✉s ❝♦♥s✐✲ ❞ér♦♥s •❡ ❝❛s ♦ù ❝❤❛q✉❡ ♠♦❞è•❡ ❞é♣❡♥❞ ❞✬✉♥ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉ ✉♥✐❞✐♠❡♥s✐♦♥✲ ♥❡• ❡st✐♠é ♣❛r •✬❊❉▼✳ ◆♦tr❡ ♦❜•❡❝t✐❢ ❡st •❛ ❝♦♥str✉❝t✐♦♥ ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ✏❛s②♠♣t♦t✐❝❛••② ❞✐str✐❜✉t✐♦♥ ❢r❡❡✑ ❞❛♥s •❡ ❝❛s ❞❡ ❝❡tt❡ ❤②♣♦t❤ès❡ ♣❛r❛♠étr✐q✉❡ ♣♦✉r •❡s ♠♦❞è•❡s tr❛✐tés✳ ▲❛ ♣r♦❜•é♠❛t✐q✉❡ ❣é♥ér❛•❡ ❞❡ ❝❡tt❡ t❤ès❡ ♣♦rt❡ s✉r •❛ ❝♦♥str✉❝t✐♦♥ ❞✬✉♥❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ s♣é❝✐❛•❡✳ ❊♥ ♣❛rt✐❝✉•✐❡r✱ ❡••❡ ❡st ♦❜t❡✲ ♥✉❡ à •✬❛✐❞❡ ❞❡ •❛ s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ❞❡ ❋r❡❞❤♦•♠ ❞✉ s❡❝♦♥❞ t②♣❡ ❛✈❡❝ •❡ ♥♦②❛✉ ❞é❣é♥éré✳ ❊♥s✉✐t❡✱ ♥♦✉s ♠♦♥tr♦♥s q✉❡ •✬❛♣♣•✐❝❛t✐♦♥ ❞❡ ❝❡tt❡ tr❛♥s❢♦r♠❛t✐♦♥ ❛✉① st❛t✐st✐q✉❡s ❞❡ ❜❛s❡ ét✉❞✐é❡s ❞❛♥s ❝❤❛q✉❡ ♠♦❞è•❡ ♥♦✉s ♣❡r♠❡t ❞✬✐♥tr♦❞✉✐r❡ ❞❡s ■◆❚|❖❉❯❈❚■❖◆ ✶✶ st❛t✐st✐q✉❡s ❛②❛♥t •❛ ♠ê♠❡ •✐♠✐t❡ ✭•✬✐♥té❣r❛•❡ ❞✉ ❝❛rré❡ ❞✉ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✮✳ ❈❡tt❡ ❞❡r♥✐èr❡ ❡st ✏❞✐str✐❜✉t✐♦♥ ❢r❡❡✑ ✈✉ q✉✬❡••❡ ♥❡ ❞é♣❡♥❞ ♥✐ ❞✉ ♠♦❞è•❡ ♥✐ ❞✉ ♣❛r❛♠ètr❡ ✐♥❝♦♥♥✉✳ ❊♥ ❡✛❡t✱ ♥♦✉s ❝♦♥str✉✐s♦♥s ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❡♥ s❡ ❜❛s❛♥t s✉r •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ s♣é❝✐❛•❡ L [•]✳ ❈❡tt❡ ♠ét❤♦❞❡ ❛ été ❛♣♣•✐q✉é❡ ♣♦✉r •❡s ♣r♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑ ❡t ❡r❣♦❞✐q✉❡ ❡t ♣♦✉r •❡ ♣r♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡ ♦❜s❡r✈és ❡♥ t❡♠♣s ❝♦♥t✐♥✉✳ ▲❛ ❝♦♥str✉❝t✐♦♥ ❞❡ L[•] ❡st ❜❛sé❡ s✉r •❛ s♦•✉t✐♦♥ ❞❡ •✬éq✉❛t✐♦♥ ❞❡ ❋r❡❞❤♦•♠ ❞✉ s❡❝♦♥❞ t②♣❡ ❛✈❡❝ •❡ ♥♦②❛✉ ❞é❣é♥éré ✭•❡s ét❛♣❡s ❞❡ ❝❡tt❡ ❝♦♥str✉❝t✐♦♥ s♦♥t ❞❛♥s ❙❡❝t✐♦♥ ✶✳✹✮✳ P♦✉r ❝❡s tr♦✐s ♠♦❞è•❡s✱ ♥♦✉s ♠♦♥tr♦♥s q✉❡ •❡s •✐♠✐t❡s ❞❡s st❛t✐st✐q✉❡s ❞❡ ❜❛s❡ ❝♦rr❡s♣♦♥❞❛♥t❡s ❛❞♠❡tt❡♥t •❛ ♠ê♠❡ r❡♣rés❡♥t❛t✐♦♥ U (ν) = W (ν) -1 0 g (ϑ, r) dW (r) ν 0 h (ϑ, r) dr,

  r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r✳ P❛r ❝♦♥séq✉❡♥t✱ •✬❛♣♣•✐❝❛t✐♦♥ ❞❡ ❝❡tt❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❛✉① st❛t✐st✐q✉❡s ❞❡ ❜❛s❡ ✐♥✐t✐❛•❡ ♥♦✉s ♣❡r♠❡t ❞❡ ❝♦♥str✉✐r❡ ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ❆❉❋ ♣♦✉r ❝❤❛q✉❡ ♠♦❞è•❡ ét✉❞✐é✳ ▼❛✐♥t❡♥❛♥t✱ ♥♦✉s ❝♦♥s✐❞ér♦♥s •❡s tr♦✐s ♠♦❞è•❡s ❞❡s ♣r♦❝❡ss✉s st♦❝❤❛st✐q✉❡s ♦❜s❡r✈és ❡♥ t❡♠♣s ❝♦♥t✐♥✉ ✿ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑✱ ❞✐✛✉s✐♦♥ ❡r❣♦❞✐q✉❡ ❡t •❡ ♣r♦❝❡ss✉s ❞❡ P♦✐ss♦♥ ♥♦♥ ❤♦♠♦❣è♥❡✳ ❙♦✉s •✬❤②♣♦t❤ès❡ ❞❡ ❜❛s❡ H 0 ✭♣❛r❛♠étr✐q✉❡✮✱ ♥♦✉s ❛✈♦♥s •❡s ♣r♦❝❡ss✉s s✉✐✈❛♥ts ✿

  a, b) ❡st ✉♥✐❞✐♠❡♥s✐♦♥♥❡• ❡t ♥♦✉s ♣r♦♣♦s♦♥s •✬❊❉▼ ♣♦✉r ❝❡ ♣❛r❛♠ètr❡ ♣♦✉r t♦✉s •❡s ♣r♦❝❡ss✉s ❞é✜♥✐s ♣ré❝é❞❡♠♠❡♥t✳ ▲❡ ❜✉t ♣r✐♥❝✐♣❛• ❡st •❛ ❝♦♥str✉❝t✐♦♥ ❞❡s t❡sts ❞✬❛•✉st❡♠❡♥t ❆❉❋ ❞❡ ♥✐✈❡❛✉ ❛s②♠♣t♦t✐q✉❡ α ∈ (0, 1) ♣♦✉r ❝❤❛q✉❡ ♠♦❞è•❡ ❞✬♦❜s❡r✈❛t✐♦♥s✳ ❈❡s t❡sts s♦♥t ❝♦♥str✉✐ts ❡♥ ❞❡✉① ét❛♣❡s✳ ❊♥ ♣r❡♠✐❡r •✐❡✉✱ ♥♦✉s ♣r♦♣♦s♦♥s ❞❡s ♣r❡♠✐èr❡s tr❛♥s❢♦r♠❛t✐♦♥s ♣♦✉r •❡s •✐♠✐t❡s ❞❡s st❛t✐st✐q✉❡s ❞❡ ❜❛s❡ r❡s♣❡❝t✐✈❡♠❡♥t ♣♦✉r ❝❤❛q✉❡ ♠♦❞è•❡✳ ❆✐♥s✐✱ ♥♦✉s ♦❜t❡♥♦♥s •❡ ♣r♦❝❡ss✉s U (•) ❛❞♠❡tt❛♥t •❛ ♠ê♠❡ r❡♣rés❡♥t❛t✐♦♥ ✭✽✮ ♣♦✉r •❡s tr♦✐s ♣r♦❝❡ss✉s ét✉❞✐és✳ ❊♥ s❡❝♦♥❞ •✐❡✉✱ ♥♦✉s ♠♦♥tr♦♥s q✉❡ •❡s st❛t✐st✐q✉❡s ❝♦♥s✐❞éré❡s ♦♥t •❛ ♠ê♠❡ •✐♠✐t❡ ✿ • Pr♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑ •♦rsq✉❡ ε → 0 ∆ε =⇒ 1 0

0 L

 0 ❞✐✛ér❡♥t❡s ♣♦✉r ❝❤❛q✉❡ ♠♦❞è•❡ ét✉❞✐é✳ ▲❛ ❞❡✉①✐è♠❡ ét❛♣❡ q✉✐ ❡st ❣é♥ér❛•❡ ♣♦✉r •❡s tr♦✐s ♠♦❞è•❡s s✬❛❣✐t ❞✬❛♣♣•✐q✉❡r ✉♥❡ ❞❡✉①✐è♠❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ L [•] ✭✶✳✷✻✮ ❛✉ ♣r♦❝❡ss✉s •✐♠✐t❡ U (•) ❡t ♦❜t❡♥✐r ✉♥ ♣r♦❝❡ss✉s ❞❡ ❲✐❡♥❡r t❡• q✉❡ 1 [U ] (ν) 2 ❞ν = ♦♣r✐été ♥♦✉s ♣❡r♠❡t ❞✬✐♥tr♦❞✉✐r❡ •❡s ✏✈❡rs✐♦♥s ❡♠♣✐r✐q✉❡s✑ ❞❡s st❛t✐s✲ t✐q✉❡s ❛❞♠❡tt❛♥t •❛ ♠ê♠❡ •✐♠✐t❡ ✭✾✮✳ P❛r ❝♦♥séq✉❡♥t✱ •❡s t❡sts ❞✬❛•✉st❡♠❡♥t ❜❛sés s✉r ❝❡s st❛t✐st✐q✉❡s s♦♥t ❆❉❋✳ ◆♦✉s ❛❜♦r❞♦♥s ❡♥s✉✐t❡ ❡♥ ❞ét❛✐•s •❡ ❝♦♥t❡♥✉ ❞❡s tr♦✐s ❝❤❛♣✐tr❡s ❞❡ ❝❡tt❡ t❤ès❡✳ ❉❛♥s ❈❤❛♣✐tr❡ ✶✱ ♥♦✉s tr❛✐t♦♥s ❞❡✉① ♣r♦❜•è♠❡s✳ ▲❡ ♣r❡♠✐❡r ❡st •❛ ❝♦♥str✉❝t✐♦♥ ❞✬✉♥❡ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ s♣é❝✐❛•❡✳ ❚♦✉s •❡s ét❛♣❡s ❞❡ ❝❡tt❡ ❝♦♥str✉❝t✐♦♥ s♦♥t ❞é✈❡•♦♣♣é❡s ❞❛♥s ❙❡❝t✐♦♥ ✶✳✹✳ ◆♦✉s ❝♦♥s✐❞ér♦♥s ❡♥s✉✐t❡ •❡ ❞❡✉①✐è♠❡ ♣r♦❜•è♠❡ ❞❛♥s ❙❡❝t✐♦♥ ✶✳✺ ✿ •❛ ❝♦♥str✉❝t✐♦♥ ❞❡ t❡st ❞✬❛•✉st❡♠❡♥t ❞❛♥s •❡ ❝❛s ❞✉ ♣r♦❝❡ss✉s ❞❡ ❞✐✛✉s✐♦♥ ❛✈❡❝ ✏♣❡t✐t ❜r✉✐t✑ ♦❜s❡r✈é ❡♥ t❡♠♣s ❝♦♥t✐♥✉✳ ◆♦✉s ♣r♦♣♦s♦♥s ✉♥ t❡st❜❛sé s✉r •❛ tr❛♥s❢♦r♠❛t✐♦♥ •✐♥é❛✐r❡ ❞é✜♥✐❡ ❞❛♥s ❚❤é♦rè♠❡ ✶✳✹✳✶ ❡t ♥♦✉s ♠♦♥tr♦♥s q✉✬✐• ❡st ❆❉❋✳ ❙✉

  s q✉❡ ❝❡tt❡ st❛✲ t✐st✐q✉❡ ❛ été ♣r♦♣♦sé❡ ♣❛r ❑✉t♦②❛♥ts ❬✹✷( ❞❛♥s •❡ ❝❛s ❞❡ •✬❊▼❱✳ ◆♦✉s ♠♦♥tr♦♥s ✶✹ ■◆❚|❖❉❯❈❚■❖◆ q✉✬❡••❡ ❝♦♥✈❡r❣❡ ❡♥ •♦✐ s♦✉s H 0 ❡t s♦✉s ❝❡rt❛✐♥❡s ❝♦♥❞✐t✐♦♥s ❞❡ ré❣✉•❛r✐tés ✈❡rs •❡

  st✉❞✐❡❞ ✐♥ ❬✸✵(-❬✸✹(✳ ❚❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡❞ ❛ •✐♥❡❛r tr❛♥s❢♦r✲ ♠❛t✐♦♥✱ ✇❤✐❝❤ ②✐❡•❞s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❡st st❛t✐st✐❝ t♦ t❤❡ ✐♥t❡❣r❛• ♦❢ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡r❡❢♦r❡ t❤❡② s❤♦✇❡❞ t❤❛t t❤❡ t❡st ❜❛s❡❞ ♦♥ t❤✐s st❛t✐st✐❝ ✐s ❆❉❋✳ ❚♦

  , τ ) λ( θn , r) ❞r, ✇❤❡r❡ Ũn ( θn , r) = √ n Λn (r) -Λ( θn , r) Λ( θn , τ )

E+TT

  ϑ |W * * T (x 1 ) -W * * T (x 2 )| 2 ≤ 3 E ϑ |ξ T (ϑ * T , x 1 )ξ T (ϑ * T , x 2 )| 2 , y, X s )σ(X s ) 2 ❞s ❞F (ϑ *

≤+

  r ❛♥② ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ l (•)✱ ✇❡ ✉s❡ t❤❡ ❢♦••♦✇✐♥❣ r❡•❛t✐♦♥ ✭❝❛♥ ❜❡ ❢♦✉fT (x) ❞x, ✇❤❡r❡ fT (x) ✐s t❤❡ •♦❝❛• t✐♠❡ ❡st✐♠❛t♦r ❣✐✈❡♥ ❜② ✭✸✳✾✮ ♦❢ t❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② f (ϑ, x)✳ |❡♠✐♥❞ t❤❛t ❜② t❤❡ ❝♦♥❞✐t✐♦♥s A 0 ❛♥❞ R 5 ✱ t❤❡ ❢✉♥❝t✐♦♥s Ṡ(ϑ, •) ❛♥❞ σ (•) ±1 ❤❛✈❡ ♣♦•②♥♦♠✐❛• ♠❛•♦r❛♥ts P (y)✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② ❤❛s ❡①♣♦✲ ♥❡♥t✐❛••② ❞❡❝r❡❛s✐♥❣ t❛✐•s ❜② ❝♦♥❞✐t✐♦♥ A 0 ✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐st t❤❡ ❝♦♥st❛♥ts C 1 > 0 ❛♥❞ λ > 0 s✉❝❤ t❤❛t f (ϑ, x) ≤ C 1 e -λ|x| . ❚❤❡r❡❢♦r❡✱ E ϑ |ξ T (ϑ * T , x 1 )ξ T (ϑ * T , x 2 )| 2 ≤ C |x 1x 2 | + C |x 1 -* T , y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y) y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y) φ1 (ϑ, y) T 0 R(ϑ, y, X s ) ❞W s ❞F (ϑ, y) ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•y, X s )❞W s ❞F (ϑ, y) , y, X s ) ❞W s . ❆❞❞✐t✐♦♥❛••②✱ ❜② ❋✉❜✐♥✐✬s t❤❡♦r❡♠ ❛♥❞ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛•✐t②✱ y, X s ) ❞W s ❞F (ϑ, y) y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) y, z) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) 2 fT (z) ❞z ≤ [F (ϑ, x 1 ) -F (ϑ, x 2 )] y, z) φ+ 2 (ϑ, y) φ1 (ϑ, y) 2 f (ϑ, z) ❞z ❞F (ϑ, y). ❚❤❡r❡❢♦r❡✱ ✉s✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s A 0 ❛♥❞ R 5 ✱ ✇❡ ♦❜t❛✐♥ y, X s ) ❞W s ❞F (ϑ, y) 2 C |x 1x 2 | . y, X s ) ❞W s ❞F (ϑ, y) , y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) ❞W s y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y)❞W s y, X s ) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) y, z) φ+ 2 (ϑ, y) φ1 (ϑ, y) ❞F (ϑ, y) ϑ, x 1 ) -F (ϑ, x 2 )] T , y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y) 2 ≤ C |x 1x 2 | . ◆♦✇✱ ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✸✳✹✸✮✳ ■♥❞❡❡❞✱ E ϑ |W * * T (x)| 2 ≤ 3 E ϑ |ξ T (ϑ * T , x)| 2 ϑ * T , y, X s )S(ϑ * T , X s ) + 1 2 Q ′ x (ϑ * T , y, X s )σ(X s ) 2 ]❞s ❞F (ϑ * T , y) ♦✈❡r✱ ✉s✐♥❣ t❤❡ s❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ❛❜♦✈❡✱ ✇❡ ❤❛✈❡ T , y, X s )σ(X s ) 2 ]❞s ❞F (ϑ * T , y) T , y, X s )σ(X s ) 2 ❞s ❞F (ϑ * T , y) 2 ≤ C. ❋✐♥❛••②✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✭✸✳✸✼✮ ✐s ♣r♦✈❡❞ ❛♥❞ t❤❡ t❡st ψ * T ✐s ❆❉❋✳ ✶✶✵ ❈❍❆P❚❊| ✸✳ ❖◆ •❖❋ ❚❊❙❚ ❋❖| ❊|•❖❉■❈ ❉■❋❋❯❙■❖◆ P|❖❈❊❙❙ ❈♦♥❝•✉s✐♦♥ ❚❤❡ ♠❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❣♦♦❞♥❡ss✲♦❢✲✜t ✭•♦❋✮ t❡sts ❢♦r t❤❡ ♣❛r❛♠❡tr✐❝ ♥✉•• ❤②♣♦t❤❡s❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ s♦♠❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ♦❜s❡r✈❡❞ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ❝♦♥s✐❞❡r t❤r❡❡ ♠♦❞❡•s ♦❢ ♦❜s❡r✈❛✲ t✐♦♥s ✿ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss ✇✐t❤ ✏s♠❛•• ♥♦✐s❡✑ X ε = (X t , 0 ≤ t ≤ T ) (ε → 0)✱ ❡r❣♦❞✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss X T = {X t , 0 ≤ t ≤ T } (T → ∞) ❛♥❞ ✐♥❤♦♠♦❣❡♥❡♦✉s τ ✲♣❡r✐♦❞✐❝ P♦✐ss♦♥ ♣r♦❝❡ss X n = X t , 0 ≤ t ≤ T = nτ (n → ∞)✳ ❯♥❞❡r t❤❡ ❜❛s✐❝ ❤②♣♦✲ t❤❡s❡s✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♠♦❞❡•s ❞❡♣❡♥❞ ♦♥ s♦♠❡ ✉♥❦♥♦✇♥ ♦♥❡✲❞✐♠❡♥s✐♦♥❛• ♣❛r❛♠❡t❡r✳ ❋♦r ❛•• t❤❡s❡ ♣r♦❝❡ss❡s✱ ✇❡ ❝♦♥str✉❝t ✏❛s②♠♣t♦t✐❝❛••② ❞✐str✐❜✉t✐♦♥ ❢r❡❡✑ ✭❆❉❋✮ •♦❋ t❡sts ψ ε ✱ ψ * n ✱ φn ❛♥❞ ψ * T ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1) ❜❛s❡❞ ♦♥ t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ L [•]✳ ❚❤❡ ❢♦r♠ ♦❢ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ ❝♦♥tr✐✲ ❜✉t✐♦♥ ♦❢ t❤✐s ✇♦r❦✳ ■t ✐s ❜❛s❡❞ ♦♥ t❤❡ s♦•✉t✐♦♥ ♦❢ t❤❡ ❋r❡❞❤♦•♠ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ✇✐t❤ ❞❡❣❡♥❡r❛t❡❞ ❦❡r♥❡•✳ ❚❤❡ t❡sts ❛r❡ ❝♦♥str✉❝t❡❞ ✐♥ t✇♦ st❡♣s✳ ❋✐rst✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❜❛s✐❝ st❛t✐st✐❝s u ε (•)✱ u * n (•) ❛♥❞ η * T (•, •) ❛♥❞ ✇❡ ✉s❡ t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r ❢♦r t❤❡ ✉♥❦✲ ♥♦✇♥ ♣❛r❛♠❡t❡r✳ ❚❤❡♥ ✇❡ s❤♦✇ t❤❛t t❤❡s❡ st❛t✐st✐❝s ❝♦♥✈❡r❣❡ t♦ t❤❡ ♣r♦❝❡ss❡s (u (t) , 0 ≤ t ≤ T )✱ (u (t) , 0 ≤ t ≤ τ ) ❛♥❞ (η (ϑ, x) , x ∈ R)✱ r❡s♣❡❝t✐✈❡•②✳ ❚❤❡♥✱ ✇❡ ♣r♦♣♦s❡ t❤❡ ✭✜rst✮ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡s❡ •✐♠✐ts ✇❤✐❝❤ •❡❛❞ t♦ t❤❡ s❛♠❡ •❛✉s✲ s✐❛♥ ♣r♦❝❡ss (0 ≤ ν ≤ 1) s) 2 ds = 1. ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r ✐♥ ❙❡❝t✐♦♥ ✷✳✸ t❤❡ ❡♠♣✐r✐❝❛• ❜❛s✐❝ st❛t✐st✐❝ U n (•, •) ✇❤✐❝❤ ❛❞♠✐ts t❤❡ s❛♠❡ •✐♠✐t U (•) ❛♥❞ t❤✐s s❤♦✇s t❤❡ ✉♥✐✈❡rs❛•✐t② ♦❢ t❤✐s •✐♠✐t ♣r♦❝❡ss✳ ◆♦t❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s h (•, •) ❛♥❞ g (•, •) ✐♥ ❞✐✛❡r❡♥t ♠♦❞❡•s ❛r❡ ❞✐✛❡r❡♥t✳ ✶✶✶ ✶✶✷ ❈❖◆❈▲❯❙■❖◆ ❚❤❡ s❡❝♦♥❞ st❡♣ ✭❣❡♥❡r❛• ❢♦r ❛•• ♠♦❞❡•s✮ ✐s t♦ ❛♣♣•② t❤❡ ✭s❡❝♦♥❞✮ •✐♥❡❛r tr❛♥s✲ ❢♦r♠❛t✐♦♥ L [•] t♦ t❤❡ ♣r♦❝❡ss U (•) s✉❝❤ t❤❛t L[U ](ν) = w ν ✱ ✇❤❡r❡ w ν , 0 ≤ ν ≤ 1 ✐s ❛ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❇❛s❡❞ ♦♥ t❤❡ ❛❜♦✈❡ ♣r♦♣❡rt②✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ✏❡♠♣✐r✐❝❛• ✈❡rs✐♦♥s✑ U ε (•)✱ U n (•, •)✱ U * n (•) ❛♥❞ ξ T (•, •) ♦❢ t❤❡ ♣r♦❝❡ss U (•) ❛♥❞ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛t✐st✐❝s t♦ t❤❡ s❛♠❡ •✐♠✐t ✿ ❚❤❡r❡❢♦r❡ ✇❡ ♣r♦♣♦s❡ t❤❡ ❆❉❋ •♦❋ t❡sts ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ∈ (0, 1) ❢♦r ❡❛❝❤ st✉❞✐❡❞ ♠♦❞❡• ψ ε = ✶ ■ {∆ε>cα} , ψ * n = ✶ ■ { ∆ * n >cα} , φn = ✶ ■ { δn>cα} , ψ * T = ✶ ■ {δ * T >cα} , ✇❤❡r❡✱ P( ∆ > c α ) = α. ❲❡ ❡♠♣❤❛s✐③❡ t❤❛t t❤❡ ❢♦✉♥❞❡❞ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ t❤✐s ✇♦r❦ ✐s r❛t❤❡r ❝✉♠❜❡rs♦♠❡ ❛♥❞ ✐ts ❛♣♣•✐❝❛t✐♦♥ ✐♥ r❡❛• ♣r♦❜•❡♠s ❝❛♥ ❜❡ ♥♦t s♦ ❡❛s②✳❆t t❤❡ s❛♠❡ t✐♠❡✱ ✇❡ ✉♥❞❡rst❛♥❞ t❤❛t t❤✐s r❡s✉•t ✐s ✐♥ s♦♠❡ s❡♥s❡ ✏♥❡❣❛t✐✈❡✑ ❛♥❞ s❛②s t❤❛t ✐❢ ✇❡ ❤❛✈❡ ♥♦ ▼▲❊ ✐t ✐s ❜❡tt❡r t♦ s❡❡❦ ❛♥♦t❤❡r •♦❋ t❡sts✱ ✇❤✐❝❤ ❛r❡ ❆❉❋✳ ◆❡✈❡rt❤❡•❡ss t❤✐s st✉❞② ❤❛s t♦ ❜❡ ❞♦♥❡ t♦ ❝♦✈❡r t❤❡ ❣❛♣ ✇❤✐❝❤ ❡①✐sts ✐♥ t❤✐s✇❡••✲❦♥♦✇♥ st❛t✐st✐❝❛• ♣r♦❜•❡♠✳ ❙♦ t❤❡ ❢✉rt❤❡r ✇♦r❦ ✇✐•• ❜❡ ❝♦♥❝❡♥tr❛t❡❞ ♦♥ ❞♦✐♥❣ ♥✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s✳ ❖✉r st❛rt✐♥❣ ♣♦✐♥t ✐s ♣r♦✈✐♥❣ t❤❛t t❤❡ ♣r♦♣♦s❡❞ t❡st ψ * n ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ✐♥❤♦♠♦❣❡✲ ♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✐s ♦❢ ❛s②♠♣t♦t✐❝ s✐③❡ α ❜✉t ✇❡ ❞♦ ♥♦t ♦❜t❛✐♥ t❤❡ ❞❡s✐r❡❞ r❡s✉•t t✐•• ♥♦✇✳ ❉✉❡ t♦ t❤❡ ✉♥✐✈❡rs❛•✐t② ♦❢ t❤❡ s♣❡❝✐❛• •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✱ ✇❡ s❤❛•• ♠❡♥t✐♦♥ t❤❛t ✐t ❝❛♥ ❜❡ ❛♣♣•✐❡❞ t♦ ♠❛♥② ♦t❤❡r st❛t✐st✐❝❛• ♠♦❞❡•s t♦ ❝♦♥str✉❝t ❆❉❋ •♦❋ t❡sts✱ ❡✳❣✳✱ ♥♦♥•✐♥❡❛r t✐♠❡ s❡r✐❡s✳ ❚❤✐s ✐s ❛♥ ✐♥t❡r❡st✐♥❣ ♣r♦❜•❡♠ ❜✉t ♥♦t ②❡t ❜❡❡♥ tr❡❛t❡❞✳ ■♥ ❛❞❞✐t✐♦♥✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ t❤❡ •✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ♦t❤❡r ✭♥♦♥ ▼▲❊✮ ❡st✐♠❛t♦rs ❛♥❞ ✐♥ ♦✉r ✇♦r❦ ✇❡ ✉s❡❞ t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❡st✐♠❛t♦r t♦ ❝♦♥str✉❝t t❤❡ ❆❉❋ •♦❋ t❡sts✳ ■t ✇✐•• ❜❡ ✐♠♣♦rt❛♥t t♦ ❝♦♥str✉❝t❆❉❋ •♦❋ t❡sts ✇✐t❤ ♦t❤❡r ❡st✐♠❛t♦rs✳ On goodness-of-fit tests with parametric hypotheses for some stochastic processes Résumé Ce travail est consacré au problème de construction des tests d'ajustement dans le cas des processus stochastiques observés en temps continu. Comme modèles d'observations, nous considérons les processus de diffusion avec « petit bruit » et ergodique et le processus de Poisson non homogène. Sous l'hypothèse nulle, nous traitons le cas où chaque modèle dépend d'un paramètre inconnu unidimensionnel et nous proposons estimateur de distance minimale pour ce paramètre. Notre but est la construction des tests d'ajustement « asymptotically distribution free » (ADF) de niveau asymtotique dans le cas de cette hypothèse paramétrique pour les modèles traités. Nous montrons alors que la limite de chaque statistique étudiée ne dépend ni du modèle ni du paramètre inconnu. Les tests d'ajustement basés sur ces statistiques sont donc ADF. L'objectif principal de ce travail est la construction d'une transformation linéaire spéciale. En particulier, nous résolvons l'équation de Fredholm du second type avec le noyau dégénéré. Sa solution nous permet de construire la transformation linéaire désirée. Ensuite, nous montrons que l'application de cette transformation aux statistiques de base étudiées dans chaque modèle nous aide à introduire des statistiques ayant la même limite (l'intégrale du carrée du processus de Wiener). Cette dernière est « distribution free » vu qu'elle ne dépend ni du modèle ni du paramètre inconnu. Par conséquent, nous proposons des tests d'ajustement ADF en se basant sur cette transformation linéaire pour les processus de diffusion avec « petit bruit » et ergodique et le processus de Poisson non homogène.

0, 1)