Daria Mishchenko
email: zenitchiki@yandex.ru

Polyfunctional pronominal markers in South-Western Mande

The paper deals with the syntactic functions and morphosyntactic features of polyfunctional (PF) pronominal markers in South-Western Mande languages (SWM). This group includes 6 languages spoken in Guinea, Liberia and Sierra Leone. The central purpose of the paper is to provide a comparative analysis of PF pronominal markers in SWM. This analysis is based on the synchronous grammar descriptions and manuals on SWM languages as well as on the data collected during the author's fieldwork with Woi-Bhalaga, a Guinean dialect of Looma. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0083)". It is typical for Mande languages to have several sets of pronominal markers opposed according to their syntactic functions and/or morphological status. One of them is often PF: for example, in Goo (Southern Mande) several sets of subject pronouns are opposed to a set of PF pronouns occurring in non-subject positions [START_REF] Vydrin | Verbal construction with a progressive meaning in Goo: Transformation of non-subject pronouns to subject ones. Handout, The Third conference "Typology of Morphosyntactic Parameters[END_REF]. Traditionally, PF markers in SWM are regarded as personal pronouns, too. However, here they are not autonomous word forms but rather inseparable and intransposable bound morphemes. Moreover, the paradigms of PF markers include two suprasegmental morphemes, these expressing the 1 st and the 3 rd person singular (except for Mende, where the 1 st person singular is expressed by means of a segmental morpheme ɲá-); they have no segment exponent and manifest themselves on the following morpheme through the change of its initial consonant and tonal alternations. On this evidence, V. [START_REF] Vydrin | Ergative/Absolutive and Active/Stative alignment in West Africa: The case of Southwestern Mande[END_REF] proposes to name such elements in SWM personal prefixes. Here, this term will be avoided in order not to refer to their morphological status. The specificity of these morphemes consists in their transcategorial nature: they can be attached to a verb (1), a noun (2) or a postposition (3):

LOOMA

(1) Gè ká-nì. 1sg.base 3sg.pi\see-aor 'I saw him'.

(2) kɛ̀ɛ̀ 1sg.pi\father 'my father'

(3) Fòlòmò ɣà níkɛ-y wódì-zù má. Folomo 3sg.npst ref\cow-def sell-ipfv 3sg.pi\on 'Folomo is selling the cow to him'.

Particular functions which can be fulfilled by PF pronominal markers in SWM vary. The standard set includes a direct object (4), an argument of a qualitative verb (5), a possessor in the situation of inalienable possession ( 6) and an indirect object with a postposition (7): MENDE (4) Ŋgá (`)-píé lɔ̀ fóló gbí mà. 1sg.ipfv 3sf.pf-do cop ref\day every ipfv 'I do it everyday'. (Innes 1971: 150)

BANDI

(5) Í-lùkpú-ŋgɔ lè . 2sg.pf-short-qual cop 'You are not tall'. (Bandi,ms.: II,17) ZIALO ( 6) (´)-tókó-y 3sg.pf-hand-def 'his hand' (Babaev 2011: 60) KPELLE (LIB.) (7) Gáàloŋ èì kɛ ̀ ɓɛ,́ èì tɔŋ̂ pù (`)-máà. ref\chief.def 3sg.cond2 be here 3sg.cond2 ref\law put 1sg.pf-on 'If only the chief was here, he'd punish me'. (Leidenfrost & McKay 2005: 77) At the same time, there are a lot of smaller differences in the use of PF pronominal markers in SWM. Thus, for example, SWM languages vary according to whether PF markers are necessary even with a full NP present or not. For example, in Looma in an intransitive resultative construction, verbal agreement with a plural subject is facultative (8), while in Zialo it is obligatory (9): LOOMA (8) Zúnú-y-tì tí-ɓɛtɛ-ʋɛ́ /zúnú-y-tì kpɛtɛ-ʋɛ. ref\man-def-pl 3pl.pi-be.wealthy-res /ref\man-def-pl be.wealthy-res 'The men are wealthy'. ZIALO (9) Pélé fèlè-gɔ-y-tì tí-w̃ɛ-yàà lè. ref\road two-def-def-pl 3pl.pf-meet-res exi 'The two roads have been linked together'. (Babaev 2011: 129) Especially interesting cases represents the use of PF markers for marking inalienable possession on the head noun. In this case their use may depend on a whole range of factors, such as the type of a noun representing possessum, the meaning of the category of number expressed by a NP in the possessor's position, its referential status, etc. In the report, a detailed analysis of such peculiarities of PF pronominal markers in SWM will be proposed.


✉ • ❛t✐♦

f ) ≤ Cρ n f

Convergence rate of the powers of an operator.

Applications to stochastic systems

T n f -π(f ) ≤ Cρ n f ✭✶✺✮ ♦r ❜② ❞✉❛•✐t②✱ ❢♦r ❛♥② ✐♥✐t✐❛• ♠❡❛s✉r❡ µ T * n µ -π T V ≤ Cρ n µ T V ✇❤❡r❡ . T V ✐s t❤❡ t♦t❛• ✈❛r✐❛t✐♦♥ ♥♦r♠✳ ❊①❛♠♣•❡ ✷✳ ▲❡t ✉s ❝♦♥s✐❞❡r ♥♦✇ ❛ ❝❤❛✐♥ ✇✐t❤ t❤❡ ❢♦r♠ X n+1 = ϕ(X n , U n )
✇❤❡r❡ U n ✐s ❛♥ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ µ✳ ❍❡♥❝❡ T f (x) = f (ϕ(x, u))µ(du).

❚❤❡ ❢✉♥❝t✐♦♥ ϕ ✐s s✉♣♣♦s❡❞ t♦ s❛t✐s❢② ❛❞❡q✉❛t❡ ♠❡❛s✉r❛❜✐•✐t② ❛ss✉♠♣t✐♦♥s ❛♥❞ ❛ ✉♥✐❢♦r♠ ❝♦♥tr❛❝t✐♦♥ ♣r♦♣❡rt② ♦♥ t❤❡ ♠❡tr✐❝ s♣❛❝❡ (S, d)✳ ❙♣❡❝✐✜❝❛••②✱ ✇❡ ❛ss✉♠❡ t❤❛t |ϕ(x, u))ϕ(y, u)| ≤ γd(x, y) ❢♦r s♦♠❡ γ < 1 ❛♥❞ ❛•• x, y, u✳ ■♥ t❤✐s ❝❛s❡ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t ✭✶✹✮ ❛♣♣•✐❡s ✇✐t❤

f = f ∞ f = f + [f ] [f ] = sup x =y f (x) -f (y) d(x, y) . ✭✶✻✮ ■♥ ♦r❞❡r t♦ ❤❛✈❡ t❤❡ . ✲❝♦♠♣❛❝t♥❡ss ♦❢ B = {T f : f ∈ E, f ≤ 1}✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ st❛t❡ s♣❛❝❡ ✐s ❝♦♠♣❛❝t✳ ❈♦♥✈❡r❣❡♥❝❡ ✐♥ t♦t❛• ✈❛r✐❛t✐♦♥ ✇✐•• ♥♦t ❤♦•❞ ✐♥ ❣❡♥❡r❛• ✭❡✳❣✳ t❤❡ ❝❤❛✐♥ X n+1 = (X n + U n )/2✱ U n ∼ B(1, 1/2)✮✳ ❇✉t ✇❡ st✐•• ❤❛✈❡ ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ str♦♥❣❡r ♥♦r♠ f → f ✳ ✶ ❚❤❡ ♣♦✐♥t ❤❛❞ ❜❡❡♥ ❛❝t✉❛••② ✐♥tr♦❞✉❝❡❞ ♠✉❝❤ s♦♦♥❡r ❜② ❉♦❡❜•✐♥ ❛♥❞ ❋♦rt❡t ✐♥ ❬✹(✱ ❊q✳✭✷✮ ❛♥❞ ✭✸✮ ♣✳ ✶✹✸✱ ❜✉t ✐♥ ❛ ♠♦r❡ s♣❡❝✐✜❝ ❝♦♥t❡①t✳ ✸ ✶✳✷ ❚❤❡ ♣r♦❜❛❜✐•✐st✐❝ ❛♣♣r♦❛❝❤ ▲❡t ✉s ❝♦♥s✐❞❡r ❛♥ ✐rr❡❞✉❝✐❜•❡ ❛♣❡r✐♦❞✐❝ ▼❛r❦♦✈ ❝❤❛✐♥ ✇✐t❤ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ π✳ ■♥t❡r❡st✐♥❣•②✱ ✐t ❛♣♣❡❛rs t❤❛t ✐♥ ♠❛♥② s✐t✉❛t✐♦♥s✱ ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡ •✐❦❡ ✭✶✺✮ ❞♦❡s ♥♦t ♦❝❝✉r✱ ❜✉t ♥❡✈❡rt❤❡•❡ss ❢♦r ♠❛♥② f ∈ E✱ T n f -π(f ) ❝♦♥✈❡r❣❡s ❡①♣♦♥❡♥t✐❛••② ❢❛st t♦ ✵✳ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐s ♥♦t . ✲✉♥✐❢♦r♠✱ ❛♥❞ s♦♠❡t✐♠❡s t❤✐s ❝♦♥✈❡r❣❡♥❝❡ ❞♦❡s ♥♦t ❢♦••♦✇ ❛♥ ❡①♣♦♥❡♥
❛♥❞ ❛ ❞r✐❢t ❝♦♥❞✐t✐♦♥ ❢♦r ❝♦♥tr♦••✐♥❣ ♠♦♠❡♥ts ♦❢ r❡❝✉rr❡♥❝❡ t✐♠❡s✳ ❆ s✐♠♣•❡ ✐••✉str❛t✐✈❡ ❡①❛♠♣•❡ ♦❢ t❤✐s ❛❜s❡♥❝❡ ♦❢ s♣❡❝tr❛• ❣❛♣ ✐s t❤❡ ❢♦••♦✇✐♥❣ ♦♣❡r❛t♦r ♦♥ (R N , . ∞ )✿ T f (x) = 1 2 (f (x) + f ((x -1) + ), x ∈ N. ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❢♦••♦✇✐♥❣ ❝❤❛✐♥ ♦♥ N X n+1 = (X n -U n+1 ) + , P (U n = 0) = P (U n = 1) = 1 2 . ❚❤❡ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ T n f (x) → π(f ) = f (0) ✐s ✈❡r② ❢❛st✱ ❜✉t t❤✐s ❝♦♥✈❡r❣❡♥❝❡ ✐s ♥♦t ✉♥✐❢♦r♠✳ ■♥ ♣❛rt✐❝✉•❛r t❤✐s ♠❛❦❡s ✭✶✺✮ ✐♠♣♦ss✐❜•❡ t♦ ♦❝❝✉r ✇✐t❤ f = f ∞ ✳ ❆ ♣♦ss✐❜•❡ ♦♣❡r❛t♦r t❤❡♦r❡t✐❝❛• ❛♣♣r♦❛❝❤ ✐s t❤❛t ♦♥❡ ❤❛s ❢♦r s♦♠❡ ✇❡❛❦❡r ♥♦r♠ . T n f -π(f ) < ρ n f ✭✶✼✮ ❢♦r ❛♥② f ∈ E✱ ❛♥❞ s♦♠❡ ✜①❡❞ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ρ n ✳ ❚❤❡ ♥♦r♠ . ✐♥tr♦❞✉❝❡❞ ❤❡r❡ ❤❛s ❛❝t✉❛••② str♦♥❣ ❝♦♥♥❡❝t✐♦♥s ✇✐t❤ t❤❡ ♦♥❡ ✐♥✈♦•✈❡❞ ✐♥ t❤❡ ■♦♥❡s❝✉ ❚✉•❝❡❛✲▼❛r✐♥❡s❝✉ ❛♣♣r♦❛❝❤✳ ❚❤❡ r❛t❡ ♦❢ ❞❡❝r❡❛s❡ ♦❢ ρ n ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ . ✳ ◆♦t✐❝❡ t❤❛t ✐❢ ✐♥ ✭✶✼✮ t❤❡ ♥♦r♠s ✇❡r❡ ❡q✉❛•✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ρ n t♦ ③❡r♦ ✇♦✉•❞ ✐♠♣•② t❤❡ ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡❀ ❤♦
Q n f ≤ ρ n f , ρ n → 0 ✭✶✽✮ ✐♥st❡❛❞ ♦❢ ✭✶✷✮✳ ❚❤✐s ✐s t❤❡ ♠❛✐♥ ♦❜•❡❝t✐✈❡ ♦❢ ❙❡❝t✐♦♥ ✷ ✭s❡❡ ❚❤❡♦r❡♠ ✶✮✳ ◆♦t✐❝❡ t❤❛t ✐♥ t❤✐s ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ✐s (E, . )✱ ❛♥❞ t❤❡ ♥♦r♠ . ♦♥•② ❛♣♣❡❛rs ✐♥ ✭✶✽✮❀ ✐♥ ♣❛rt✐❝✉•❛r ♥♦t❤✐♥❣ ❣✉❛r❛♥t❡❡s t❤❛t Q n f t❡♥❞s t♦ ③❡r♦✳ ❙❡❝t✐♦♥ ✸ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡✱ ✐✳❡✳ ρ n = Cρ n ✳ ❙♣❡❝✐✜❝❛••② ❚❤❡♦r❡♠ ✸ s❤♦✇s ❤♦✇ t❤❡ (♦s❤✐❞❛✲❑❛❦✉t❛♥✐ ❛♥❞ ■♦♥❡s❝✉ ❚✉•❝❡❛✲▼❛r✐♥❡s❝✉ ❛♣♣r♦❛❝❤❡s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ ❛ s✐♥❣•❡ st❛t❡♠❡♥t✳ ❚❤✐s ❛••♦✇s ❛♥ ❡❛s② tr❡❛t♠❡♥t ♦❢ ❝❤❛✐♥s ❤❛✈✐♥❣ ❛♥ ✐rr❡❞✉❝✐❜•❡ ❝♦♠♣♦♥❡♥t ❛♥❞ ❛♥♦t❤❡r ❝♦♠♣♦♥❡♥t ❜❡❤❛✈✐♥❣ •✐❦❡ ❊①❛♠♣•❡ ✷ ❛❜♦✈❡✳ ❙❡❝t✐♦♥ ✹ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ s✉❜✲❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤❡♦r❡♠ ✼ ♣r♦♣♦s❡s ❛ ✇❛② t♦ ❡st✐♠❛t❡ t❤❡ ❞❡❝❛② r❛t❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ ρ n ✳ •❡♥❡r❛• t❤❡♦r❡♠s ❝♦♥❝❡r♥✐♥❣ ▼❛r❦♦✈ ❝❤❛✐♥s ❛♥❞ ❡①❛♠♣•❡s ❛r❡ ❣✐✈❡♥ t❤r♦✉❣❤♦✉t t❤❡ ♣❛♣❡r ✐♥ ♦r❞❡r t♦ ♣♦✐♥t ♦✉t t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ✐s ✈❡r② ✈❡rs❛t✐•❡ ❢♦r t❤❡ st✉❞② ♦❢ ❛ •❛r❣❡ ❝•❛ss ♦❢ ❞②♥❛♠✐❝❛• s②st❡♠s✱ ✐♥ ♣❛rt✐❝✉•❛r ❢♦r ✐rr❡❞✉❝✐❜•❡ ❛s ✇❡•• ❛s ❢♦r ♥♦♥✲✐rr❡❞✉❝✐❜•❡ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ✹ ✷ •❡♥❡r❛• r❡s✉•ts ■♥ t❤❡ ✇❤♦•❡ ♣❛♣❡r✱ ✇❡ s❤❛•• ❝♦♥s✐❞❡r ❛♥ ♦♣❡r❛t♦r T ♦♥ ❛ ✈❡❝t♦r s♣❛❝❡ (E, . ) ❡♥❞♦✇❡❞ ✇✐t❤ ❛♥♦t❤❡r ♥♦r♠ . ✳ ❲❡ s❤❛•• ❞❡♥♦t❡ ❜② B 0 ✱ B t❤❡ ✉♥✐t ❜❛••s ❢♦r t❤❡s❡ ♥♦r♠s✿ B 0 = {f ∈ E : f ≤ 1}, ✭✶✾✮ B = {f ∈ E : f ≤ 1}. ✭✷✵✮ ❲❡ s❤❛•• ✇♦r❦ ✉♥❞❡r t❤❡ ❢♦••♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s ✭❆✵✮ (E, . ) ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ B ✐s ❝♦♠♣•❡t❡ ❢♦r t❤❡ ♠❡tr✐❝ ✐♥❞✉❝❡❞ ❜② . ✱ ❛♥❞ ❢♦r s♦♠❡ C 0 ∀f ∈ E, f ≤ C 0 f . ✭✷✶✮ ✭❆✶✮ ❚❤❡ ♥✉♠❜❡r C T = sup n T n ✐s ✜♥✐t❡✳ (E, . ) ✐s t②♣✐❝❛••② ♥♦t ❝♦♠♣•❡t❡✳ ❋♦r ✐♥st❛♥❝❡ ♦♥❡ ❝❛♥ ❤❛✈❡ E = C b (R)✱ f = f ∞ ❛♥❞ f = sup x |f (x)| 1+|x| 2 ✳ ❚❤❡♦r❡♠ ✶✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✭❆✵✮ ❛♥❞ ✭❆✶✮✱ T ✐s ❛ s✉♠ ♦❢ t✇♦ ♦♣❡r❛t♦rs T = K + V ✭✷✷✮ ❜♦t❤ . ✲❝♦♥t✐♥✉♦✉s ❛♥❞ . ✲❝♦♥t✐♥✉♦✉s✱ ✇❤✐❝❤ s❛t✐s❢② ❢♦r s♦♠❡ C K > 0 ❛♥❞ ❢♦r ❛♥② n ❛♥❞ ❛♥② f ∈ E KT n KB ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞, ✭✷✸✮ V n f ≤ ε ′ n f , ε ′ n → 0, ✭✷✹✮ k≥0 KV k < ∞, ✭✷✺✮ Kf ≤ C K f , ✭✷✻✮ t❤❡♥ ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮ ❤♦•❞ tr✉❡✳ ■❢ T ✐s . ✲❝♦♥t✐♥✉♦✉s ❛♥❞ . ✲❝♦♥t✐♥✉♦✉s✱ ❛♥❞ T k s❛t✐s✜❡s t❤❡ ❛ss✉♠♣t✐♦♥s ❛❜♦✈❡ ❢♦r s♦♠❡ k > 0✱ t❤❡♥ ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮ ❤♦•❞ tr✉❡✳ ❚❤❡ ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❆✳ ❚❤✐s ♣r♦♦❢ ✉t✐•✐③❡s t❤❡ ♠♦r❡ ❣❡♥❡r❛• ❚❤❡♦r❡♠ ✶✵ st❛t❡❞ ✐♥ ❙❡❝t✐♦♥ ❆✳✶✱ ❛♥❞ ✐s ❜❛s❡❞ ♦♥ ❛♥ ❡①t❡♥s✐✈❡ ✉s❡ ♦❢ t❤❡ ✐❞❡♥t✐t②✿ T n = n i=1 T n-i (T -V )V i-1 + V n = n i=1 T n-i KV i-1 + V n . ✭✷✼✮ ❱❡r② ❝♦❛rs❡•② t❤❡ ❛ss✉♠♣t✐♦♥s ✐♠♣•② t❤❛t ❢♦r ❛♥② s❡q✉❡♥❝❡ f k ∈ B✱ t❤❡ s❡q✉❡♥❝❡ T k f k ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✳ ❚❤✐s ❛••♦✇s ✉s t♦ ♣r♦✈❡ t❤❛t E ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ . ✲❝•♦s❡❞✱ T ✲st❛❜•❡ s♣❛❝❡s E = {f : T n f → 0} ⊕ {f : lim inf n f -T n f = 0} = E 0 ⊕ E c . ✭✷✽✮ ◆❡①t ✇❡ ♣r♦✈❡ t❤❛t E c ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ✭❜② ♣r♦✈✐♥❣ t❤❛t ✐ts ✉♥✐t ❜❛•• ✐s ❝♦♠♣❛❝t✮ ✇✐t❤ ❛ ❜❛s✐s ♦❢ ❡✐❣❡♥✈❡❝t♦rs✳ ❚❤❡ ♣r♦•❡❝t✐♦♥ P 0 ♦❢ ❊q✉❛t✐♦♥ ✭✼✮ ✐s t❤❡♥ t❤❡ ♣r♦•❡❝t✐♦♥ ♦♥ E 0 ♣❛r❛••❡• t♦ E c ✳ ❆♣♣•✐❝❛t✐♦♥ t♦ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❲❡ s❤❛•• ❝♦♥s✐❞❡r ❛ ♠❡❛s✉r❛❜•❡ s♣❛❝❡ (S, F ) ✇✐t❤ ❛ ♠❡❛s✉r❛❜•❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ v ≥ 1 ❛♥❞ ✇❡ ❛❞♦♣t t❤❡ ❢♦••♦✇✐♥❣ ♥♦t❛t✐♦♥ f v = f /v ∞ . ✭✷✾✮ ❲❡ ❞❡♥♦t❡ ❜② E t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ ❜♦✉♥❞❡❞ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s ♦♥ (S, F )✳ ❲❡ r❡❝❛•• t❤❛t ❛ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r ♦♥ (S, F ) ✐s ❛ ❢✉♥❝t✐♦♥ (x, A) → T (x, A) s✉❝❤ t❤❛t ❢♦r ❛♥② x ∈ S✱ A → T (x, A) ✐s ❛ ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡✱ ❛♥❞ ❢♦r ❛♥② A ∈ F ✱ x → T (x, A) ✐s ♠❡❛s✉r❛❜•❡✳ ✺ ❚❤❡♦r❡♠ ✷✳ ▲❡t T ❜❡ ❛ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r✿ (T f )(x) = y f (y)T (x, dy). ❆ss✉♠❡ t❤❛t ❢♦r s♦♠❡ s❡t K 0 ❛♥❞ s♦♠❡ c v > 0 T v(x) ≤ v(x) -c v , ∀x / ∈ K 0 ✭✸✵✮ T v ✐s ❜♦✉♥❞❡❞ ♦♥ K 0 ✭✸✶✮ ❛♥❞ t❤❛t t❤❡r❡ ❡①✐sts ❛♥♦t❤❡r ❦❡r♥❡• K(x, dy) s✉❝❤ t❤❛t 0 ≤ K(x, dy) ≤ T (x, dy)✱ ❛♥❞ s✉❝❤ t❤❛t ❢♦r s♦♠❡ ε > 0✱ ❛♥❞ s♦♠❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♠❡❛s✉r❡ ν ♦♥❡ ❤❛s K(x, S) ≥ ε, ∀x ∈ K 0 ✭✸✷✮ K(x, S) = 0, ∀x / ∈ K 0 ✭✸✸✮ Kf ∞ ≤ ν(|f |), ∀f ∈ E ✭✸✹✮ ν(v) < ∞. ✭✸✺✮ ❙❡t f = f ∞ ✭✸✻✮ f = f v . ✭✸✼✮ ❚❤❡♥ ❚❤❡♦r❡♠ ✶ ❛♣♣•✐❡s ✇✐t❤ K ❛♥❞ V = T -K✳ ■♥ ♣❛rt✐❝✉•❛r ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮ ❤♦•❞ tr✉❡✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ t❤❡r❡ ✐s ♥♦ ♠❡❛s✉r❛❜•❡ s❡t A s✉❝❤ t❤❛t x → T (x, A) ✐s ❛ ♥♦♥✲tr✐✈✐❛• ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ✷ t❤❡♥ t❤❡r❡ ❡①✐st ❛ ♠❡❛s✉r❡ π ❛♥❞ ❛ s❡q✉❡♥❝❡ ρ n → 0 s✉❝❤ t❤❛t ❢♦r ❛♥② f ∈ E T n f -π(f ) v ≤ ρ n f ∞ . ✭✸✽✮ ❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❚❤❡♦r❡♠ ✶ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❇✳ |❡♠❛r❦✳ ❊q✉❛t✐♦♥ ✭✸✵✮ ✐s ❦♥♦✇♥ ❛s t❤❡ ✧❞r✐❢t ❝♦♥❞✐t✐♦♥✧ ✭❝❢ ❚❤❡♦r❡♠ ✶✶✳✵✳✶ ♦❢ ❬✶✸( ♦r Pr♦♣♦s✐t✐♦♥ ✺✳✶✵ ✐♥ ❬✶✹(✮✳ ❊q✉❛t✐♦♥s ✭✸✷✮ t♦ ✭✸✺✮ ❛r❡ r❡♠✐♥✐s❝❡♥t ♦❢ t❤❡ T ✲❝❤❛✐♥ ♣r♦♣❡rt② ✭❝❢ ❬✶✸( ❚❤❡♦r❡♠ ✻✳✵✳✶✮✱ ✉s❡❞ t♦ ❝❤❡❝❦ t❤❡ ✐rr❡❞✉❝✐❜✐•✐t② ❛ss✉♠♣t✐♦♥ ✭❝❢✳ ❬✶✸( ♣✳ ✽✼✮✳ ❍♦✇❡✈❡r✱ t❤❡ ❋❡••❡r ♣r♦♣❡rt② ✐s ♥♦t r❡q✉✐r❡❞ ❤❡r❡✳ ❊q✉❛t✐♦♥ ✭✸✸✮ ✐s ♥♦t ❛ r❡str✐❝t✐♦♥✱ s✐♥❝❡ ❝❛♥❝❡••✐♥❣ K ♦✉ts✐❞❡ K 0 ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ♦t❤❡r ❛ss✉♠♣t✐♦♥s✳ ❚❤❡ ❡ss❡♥t✐❛• ❞✐✣❝✉•t② ✇✐t❤ t❤❡ ♣r❡s❡♥t ❛ss✉♠♣t✐♦♥s ✐s t❤❛t t❤❡ s❡t K 0 ❤❛s t♦ ❜❡ t❤❡ s❛♠❡ ✐♥ ✭✸✵✮ ❛♥❞ ✐♥ ✭✸✷✮✳ ◆♦t✐❝❡ ❤♦✇❡✈❡r t❤❛t t❤❡ s❡ts K 0 s❛t✐s❢②✐♥❣ ❛ss✉♠♣t✐♦♥s ✭✸✷✮ ❛♥❞ ✭✸✸✮ ❛r❡ st❛❜•❡ ❜② ✜♥✐t❡ ✉♥✐♦♥✳ ❊①❛♠♣•❡✳ ❈♦♥s✐❞❡r t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ♦♥ R + ❞❡✜♥❡❞ ❜② X n+1 = X n + 1 + X α n W n+1 ✇❤❡r❡ W n ✐s ❛♥ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ♦❢ ♥♦♥✲③❡r♦ ❝❡♥tr❡❞ r❛♥❞♦♠ ✈❛r✐❛❜•❡s ✇✐t❤ ✈❛•✉❡s ✐♥ [-1, 1]✱ ✇✐t❤ ❛ ♥♦♥ ③❡r♦ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ❝♦♠♣♦♥❡♥t✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❛ss✉♠❡ t❤❛t 1/2 < α < 1. ❚❛❦❡ v(x) = x p + 1 ❢♦r s♦♠❡ p ≤ 1 ✇❤✐❝❤ ✇✐•• ❜❡ ❝❤♦s❡♥ •❛t❡r ❛s 2(1 -α)✳ ❚❤❡♥ T v(x) = 1 + E[(x + 1 + x α W 1 ) p ]. ✷ ❚❤✐s ✇♦✉•❞ ♠❡❛♥ t❤❛t 1 X 1 ∈A ✇♦✉•❞ ❜❡ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥✲❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛• st❛t❡ X 0 ✳ ✻ ❇② t❤❡ s❡❝♦♥❞ ♦r❞❡r ❚❛②•♦r ❢♦r♠✉•❛ ❛♣♣•✐❡❞ t♦ t❤❡ ❢✉♥❝t✐♦♥ v ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ x + 1✱ t❤❡r❡ ❡①✐st ❛ r❛♥❞♦♠ ♥✉♠❜❡r 0 < θ < 1 s✉❝❤ t❤❛t T v(x) = 1 + (x + 1) p - p(1 -p) 2 x 2α E[(x + 1 + θx α W 1 ) p-2 W 2 1 ] ≤ 1 + (x + 1) p - p(1 -p) 2 x 2α (x + 1 -x α ) p-2 σ 2 ✇❤❡r❡ σ 2 ✐s t❤❡ ✈❛r✐❛♥❝❡ ♦❢ W 1 ✳ ❚❛❦✐♥❣ p = 2(1 -α)✱ ✇❡ ❤❛✈❡ 0 < p < 1 ❛♥❞ T v(x) ≤ 1 + (x + 1) p - p(1 -p) 2 x x + 1 -x α 2α σ 2 ≤ 1 + x p - p(1 -p) 3 σ 2 ❢♦r x •❛r❣❡ ❡♥♦✉❣❤✳ ❊q✉❛t✐♦♥ ✭✸✵✮ ✐s s❛t✐s✜❡❞ ❢♦r s♦♠❡ ✐♥t❡r✈❛• K 0 = [0, M ]✳ ❊q✉❛t✐♦♥ ✭✸✶✮ ✐s ♦❜✈✐♦✉s✳ ■♥ ♦r❞❡r t♦ ❝❤❡❝❦ ❊q✉❛t✐♦♥s ✭✸✷✮ t♦ ✭✸✺✮✱ ♥♦t✐❝❡ t❤❛t ✐❢ t❤❡ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ❝♦♠♣♦♥❡♥t ♦❢ W 1 ❤❛s ❛ ❞❡♥s✐t② ≥ ε ♦♥ ❛ s✉❜s❡t A ♦❢ [-1, 1] ✇✐t❤ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✱ K(x, dy) ❝❛♥ ❜❡ t❛❦❡♥ ❛s ελ(A) t✐♠❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ x + 1 + x α W1 ✱ ✇❤❡r❡ W1 ❤❛s ❞❡♥s✐t② 1 A /λ(A)✱ ν ✐s s♦♠❡ ♠✉•t✐♣•❡ ♦❢ t❤❡ ✉♥✐❢♦r♠ ♠❡❛s✉r❡ ♦♥ [0, M + M α + 1]✳ ❚❤❡r❡❢♦r❡ t❤❡♦r❡♠ ❛♣♣•✐❡s✳ ■♥ ♦r❞❡r t♦ ❣❡t ✭✸✽✮ ✐t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t T 1 A = 1 B ✐s ✐♠♣♦ss✐❜•❡ ✉♥•❡ss B = R + ♦r B = ∅✳ ■❢ B ✐s ♥♦♥ tr✐✈✐❛• ♦♥❡ ❝❛♥ ✜♥❞ t✇♦ s❡q✉❡♥❝❡s x n ❛♥❞ y n ❤❛✈✐♥❣ t❤❡ s❛♠❡ •✐♠✐t s✉❝❤ t❤❛t x n ∈ B ❛♥❞ y n / ∈ B✳ ❚❤❡ r❡•❛t✐♦♥ T 1 A = 1 B ✇♦✉•❞ ♠❡❛♥ t❤❛t ❢♦r ❡❛❝❤ n✱ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ x n + 1 + x α n W 1 ❛♥❞ y n + 1 + y α n W 1 ❛r❡ ♠✉t✉❛••② s✐♥❣✉•❛r ✭s✉♣♣♦rt❡❞ ♦♥ A ❛♥❞ A c ✮✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜•❡ ❢♦r n •❛r❣❡ ❜❡❝❛✉s❡ W 1 ❤❛s ❛♥ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ❝♦♠♣♦♥❡♥t✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ B ✐s ♥❡❝❡ss❛r✐•② tr✐✈✐❛• ❛♥❞ ✭✸✽✮ ❤♦•❞s✳ ◆♦t✐❝❡ t❤❛t ♥❡✈❡rt❤❡•❡ss E[X n ] = E[X 0 ] + n✳ ✸ •❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡✿ ◗✉❛s✐✲❝♦♠♣❛❝t♥❡ss ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❣✐✈❡ ❛ t❤❡♦r❡♠ ✇❤✐❝❤ ❡♥❝♦♠♣❛ss❡s ❜♦t❤ (♦s✐❞❛✲❑❛❦✉t❛♥✐ ❛♥❞ ■♦♥❡s❝✉✲❚✉•❝❡❛✲▼❛r✐♥❡s❝✉ ❚❤❡♦r❡♠s✱ ❛♥❞ ♣r❡s❡♥t ❛♥ ❛♣♣•✐❝❛t✐♦♥ t♦ ▼❛r❦♦✈ ❝❤❛✐♥s ✇❤✐❝❤ ♠✐①❡s ❜♦t❤ ❦✐♥❞s ♦❢ s✐t✉❛t✐♦♥s ♣r❡s❡♥t❡❞ ❛❜♦✈❡✳ ❆ s♣❡❝✐✜❝ ❛♣♣•✐❝❛t✐♦♥ t♦ ❛✉t♦r❡❣r❡ss✐✈❡ ♣r♦❝❡ss❡s ✇✐t❤ ▼❛r❦♦✈ s✇✐t❝❤✐♥❣ ✐s ✜♥❛••② st✉❞✐❡❞✳ ❲❡ r❡❝❛•• t❤❛t B ❞❡♥♦t❡s t❤❡ ✉♥✐t ❝•♦s❡❞ ❜❛•• ❢♦r t❤❡ ♥♦r♠ . ✳ ❚❤❡♦r❡♠ ✸✳ ▲❡t T ❜❡ ❛♥ ♦♣❡r❛t♦r ♦♥ (E, . ) s❛t✐s❢②✐♥❣ ✭❆✵✮✱ ✭❆✶✮ ❛♥❞ ✭❆✷✮ T ✐s . ✲❝♦♥t✐♥✉♦✉s✳ ❋♦r s♦♠❡ . ✲t♦t❛••② ❜♦✉♥❞❡❞ s❡t K B ✱ γ < 1✱ c > 0 ❛♥❞ q > 0 T q B ⊂ γB + K B ✭✸✾✮ T q f ≤ γ f + c f . ✭✹✵✮ ❚❤❡♥ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✷✮ ❤♦•❞✳ ▲✐❦❡ ❚❤❡♦r❡♠ ✶✱ t❤✐s t❤❡♦r❡♠ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❣❡♥❡r❛• ❚❤❡♦r❡♠ ✶✵ st❛t❡❞ ✐♥ ❙❡❝t✐♦♥ ❆✳✶❀ ✐ts ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❈✳ ❚❤❡ ❢♦••♦✇✐♥❣ t❤❡♦r❡♠ ♠❛② s❡❡♠ ✈❡r② ❣❡♥❡r❛• ❛♥❞ ✉♥❝•❡❛r ❢♦r t❤❡ ❛♣♣•✐❝❛t✐♦♥s✳ ❲❡ s❤♦✉•❞ ♣♦✐♥t ♦✉t t❤❛t ✇❡ ✐♥t❡♥❞ t♦ ❜r✐❞❣❡ ❛ ❝♦♥t✐♥✉✉♠ ♦✈❡r t✇♦ ❡①tr❡♠❡ ❝❛s❡s✿ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✐♥ ❲❛ss❡rst❡✐♥ ❞✐st❛♥❝❡ ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t♦t❛• ✈❛r✐❛t✐♦♥✳ ❚❤✐s ✇✐•• ❜❡ ❡①❡♠♣•✐✜❡❞ ✐♥ ❚❤❡♦r❡♠ ✻✳ ▲❡t ✉s •✉st ♠❡♥t✐♦♥ t❤❛t [.] ❜❡•♦✇ ✐s t②♣✐❝❛••② ❛ ▲✐♣s❝❤✐t③ ♥♦r♠ •✐❦❡ ✐♥ ❊q✉❛t✐♦♥ ✭✶✻✮✱ ❛♥❞ t❤❛t t❤❡ r❡str✐❝t✐♦♥ t♦ s❡ts S x ✐♥ ✭✹✺✮ t♦ ✭✹✼✮ ❛••♦✇s t♦ ❣❡t ♠♦r❡ •♦❝❛• ❛ss✉♠♣t✐♦♥s ✭✐✳❡✳ •❡ss ✉♥✐❢♦r♠✐t②✮✳ ❚❤❡ t❤❡♦r❡♠ s❛②s t❤❛t ✐❢ •♦❝❛••② T ❝❛♥ ❜❡ •♦✇❡r ❜♦✉♥❞❡❞ ❜② ❛♥ ♦♣❡r❛t♦r ✇✐t❤ ♥✐❝❡ ♣r♦♣❡rt✐❡s✱ t❤❡♥ q✉❛s✐✲❝♦♠♣❛❝t♥❡ss ❤♦•❞s✳ ✼ ❚❤❡♦r❡♠ ✹✳ ▲❡t (S, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ B ✐ts ❇♦r❡• σ✲✜❡•❞✳ ❲❡ ❛ss✉♠❡ t❤❛t ✐s ❣✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ v(x) ≥ 1 ♦♥ S s✉❝❤ t❤❛t ❢♦r ❛♥② A > 0✱ {x : v(x) ≤ A} ✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r ❛ ✈❡❝t♦r s♣❛❝❡ E ♦❢ B✲♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ S ❝♦♥t❛✐♥✐♥❣ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s✳ ❖♥ E ✐s ❞❡✜♥❡❞ ❛ s❡♠✐✲♥♦r♠ f → [f ] ❛♥❞ ✇❡ s❡t ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ♦♥ S✿ f = f + [f ], ✭✹✶✮ f = sup x |f (x)| v(x) . ✭✹✷✮ ❲❡ ❛ss✉♠❡ t❤❛t (E, . ) ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ t❤❛t ✭❆✵✮ ❤♦•❞s✳ ▲❡t T ❜❡ ❛ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ♦♥ E✳ ❲❡ ❛ss✉♠❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ 0 < γ b , γ v < 1 ❛♥❞ c v > 0 s✉❝❤ t❤❛t [T f ] ≤ γ b [f ], f ∈ E, ✭✹✸✮ T v(x) ≤ γ v v(x) + c v . ✭✹✹✮ ❲❡ ❛ss✉♠❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ε d > 0 ❛♥❞ c d > 0✱ ❛♥❞ ❢♦r ❛♥② x ∈ S✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❦❡r♥❡• K x (y, dz) ❛♥❞ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ S x ♦❢ x✱ s✉❝❤ t❤❛t ❢♦r ❛♥② y ∈ S x ❛♥❞ f ∈ E✱ K x (y, dz) ≤ T (y, dz), ✭✹✺✮ K x (y, S) ≥ ε d (x), ✭✹✻✮ |K x f (y) -K x f (x)| ≤ c d (x, y)[f ]. ✭✹✼✮ ▼♦r❡♦✈❡r t❤❡ ❢✉♥❝t✐♦♥ c d (., .) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❜♦✉♥❞❡❞ ♦♥ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ S × S✱ ❛♥❞ ε d (x) ✐s ♣♦s✐t✐✈❡ ❛♥❞ s❛t✐s✜❡s lim v(x)→∞ ε d (x)v(x) = +∞, ✭✹✽✮ ∀ A, min v(x)≤A ε d (x) > 0. ✭✹✾✮ ❚❤❡♥ ❚❤❡♦r❡♠ ✸ ❤♦•❞s ✇✐t❤ ❛ ♣❛✐r ♦❢ ♥♦r♠s ( . ′ , . ′ ) r❡s♣❡❝t✐✈❡•② ❡q✉✐✈❛•❡♥t t♦ . ❛♥❞ . ✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ ♦♥•② ❡✐❣❡♥✈❡❝t♦rs ♦❢ T ✇✐t❤ ❛♥ ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ✶✱ t❤❡r❡ ❡①✐st C > 0✱ 0 < ρ < 1 ❛♥❞ ❛ ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡ π s✉❝❤ t❤❛t ❢♦r ❛♥② f ∈ E✱ π(f )1 -T n f ≤ Cρ n f ✭✺✵✮ ❛♥❞ π(v) < ∞✳ ❚❤❡ ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❉✳ ❲❡ ✉s❡ ❚❤❡♦r❡♠ ✸ ✇✐t❤ q = 1✳ ❚❤❡ ✐❞❡❛ ✐s t♦ s❡t Kf (x) = n i=1 θ i (x)K xi f (x i ) ✇❤❡r❡ θ 1 , . . . θ n ✐s ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✉♥✐t② ♦❢ ❛ •❛r❣❡ ♣♦rt✐♦♥ ♦❢ t❤❡ s♣❛❝❡✱ ❡❛❝❤ x i ❜❡✐♥❣ ❛ ♣♦✐♥t ♦❢ t❤❡ s✉♣♣♦rt ♦❢ θ i ✳ ❈•❡❛r•② K(B) ✐s ❝♦♠♣❛❝t✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t (T -K)f ≤ γ f ✭✇❤✐❝❤ ✐♠♣•✐❡s ✭✸✾✮✮ ❛♥❞ t❤❛t ✭✹✵✮ ❤♦•❞s tr✉❡✳ ❲❡ s❤❛•• ❝♦♥s✐❞❡r t✇♦ ❡①❛♠♣•❡s✱ ♦♥❡ ✇❤❡r❡ [f ] = 0 ❛♥❞ ✇❡ ❣❡t ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ . ♥♦r♠ ✭✇❤✐❝❤✱ ❜② ❞✉❛•✐t②✱ ❝♦rr❡s♣♦♥❞s t♦ ❣❡♦♠❡tr✐❝ ✇❡✐❣❤t❡❞ t♦t❛• ✈❛r✐❛t✐♦♥ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥✮✱ ❛♥❞ ❛♥♦t❤❡r ❝❛s❡ ✇❤❡r❡ [.] ♣•❛②s ❛♥ ✐♠♣♦rt❛♥t r♦•❡✳ ✽ ❆♣♣•✐❝❛t✐♦♥ t♦ ❣❡♦♠❡tr✐❝ t♦t❛• ✈❛r✐❛t✐♦♥ ❝♦♥✈❡r❣❡♥❝❡✳ ■♥ t❤❡ ❝❛s❡ [f ] = 0✱ t❤❡ ❦❡r♥❡• K x (y, dz) s❤♦✉•❞ ♥♦t ❞❡♣❡♥❞ ♦♥ y✱ ❛♥❞ ✇❡ ❣❡t t❤❡ ❢♦••♦✇✐♥❣ ❝♦r♦••❛r②✿ ❈♦r♦••❛r② ✺✳ ✭❲❡✐❣❤t❡❞ •♦❝❛• ❉♦❡❜•✐♥ ❝♦♥❞✐t✐♦♥✮ ▲❡t (S, d) ❜❡ ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ B ✐ts ❇♦r❡• σ✲✜❡•❞✳ ❲❡ ❛ss✉♠❡ t❤❛t ✐s ❣✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ v(x) ≥ 1 ♦♥ S s✉❝❤ t❤❛t ❢♦r ❛♥② A > 0✱ {x : v(x) ≤ A} ✐s ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r ❛ ✈❡❝t♦r s♣❛❝❡ E ♦❢ B✲♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ S ❝♦♥t❛✐♥✐♥❣ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s✳ ❲❡ s❡t ❢♦r ❛♥② ❢✉♥❝t✐♦♥ f ♦♥ S✿ f = sup x |f (x)| v(x) . ✭✺✶✮ ❲❡ ❛ss✉♠❡ t❤❛t (E, . ) ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❛♥❞ t❤❛t ✭❆✵✮ ❤♦•❞s✳ ▲❡t T ❜❡ ❛ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ♦♥ E✳ ❲❡ ❛ss✉♠❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ 0 < γ v < 1 ❛♥❞ c v > 0 s✉❝❤ t❤❛t T v(x) ≤ γ v v(x) + c v . ✭✺✷✮ ❲❡ ❛ss✉♠❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ε d > 0✱ ❛♥❞ ❢♦r ❛♥② x ∈ S ♦❢ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❦❡r♥❡• K x (dz) ❛♥❞ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ S x ♦❢ x✱ s✉❝❤ t❤❛t K x (dz) ≤ T (y, dz), y ∈ S x ✭✺✸✮ K x (S) ≥ ε d (x). ✭✺✹✮ ❚❤❡ ❢✉♥❝t✐♦♥ ε d (x) ✐s ♣♦s✐t✐✈❡ ❛♥❞ s❛t✐s✜❡s lim v(x)→∞ ε d (x)v(x) = +∞, ✭✺✺✮ ∀ A, min v(x)≤A ε d (x) > 0. ✭✺✻✮ ❚❤❡♥ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✷✮ ❤♦•❞✳ ■♥ ♣❛rt✐❝✉•❛r✱ ✐❢ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ ♦♥•② ❡✐❣❡♥✈❡❝t♦rs ♦❢ T ✇✐t❤ ❛♥ ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ✶✱ t❤❡r❡ ❡①✐st C > 0✱ 0 < ρ < 1 ❛♥❞ ❛ ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡ π s✉❝❤ t❤❛t ❢♦r ❛♥② f ∈ E✱ π(f ) -T n f ≤ Cρ n f ✭✺✼✮ ❛♥❞ π(v) < ∞✳ ■♥ ♠❛♥② ❝❛s❡s ε d (x) = 1/2 ✇✐•• ❞♦ t❤❡ •♦❜✱ ❜✉t ✐♥ t❤❡ ❢♦••♦✇✐♥❣ ❡①❛♠♣•❡ t❤❡ s✐t✉❛t✐♦♥ ✐s ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞✿ X n+1 = 1 2 X n ✇✐t❤ ♣r♦❜❛❜✐•✐t② 1 -p(X n ) V n ✇✐t❤ ♣r♦❜❛❜✐•✐t② p(X n ) ✇❤❡r❡ V n ✐s ❛♥ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ❛♥❞ p ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦❢ x❀ V n ❝❛♥ ❜❡ ❝♦♥st❛♥t✳ ❲❡ s❡❡ t❤❛t ♦♥•② t❤❡ s❡❝♦♥❞ t②♣❡ ♦❢ tr❛♥s✐t✐♦♥ ❝♦♥tr✐❜✉t❡s t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t♦t❛• ✈❛r✐❛t✐♦♥✱ ❡✈❡♥ ✐❢ V n ≡ 0✱ t❤✐s ✐s ✇❤② ✇❡ s❤❛•• ♥❡❡❞ p(x) ♥♦t t♦ ❜❡ t♦♦ s♠❛••✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t ❢♦r s♦♠❡ 0 < α < 1 E[|V n | α ] < ∞ lim x→∞ p(x)|x| α = +∞ p ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❤❡♥ ❊q✉❛t✐♦♥s ✭✺✷✮ t♦ ✭✺✻✮ ❛r❡ ❝•❡❛r•② s❛t✐s✜❡❞ ✇✐t❤ v(x) = |x| α + 1 K x (f ) = 1 2 p(x)E[f (V 1 )] ε d (x) = 1 2 p(x) S x = {y : p(y) > p(x)/2} ❛♥❞ t❤❡ ❡①♣♦♥❡♥t✐❛• ❝♦♥✈❡r❣❡♥❝❡ ❤♦•❞s✳ ◆♦t✐❝❡ t❤❛t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ p ✐s ❢❛r ❢r♦♠ ❜❡✐♥❣ ♥❡❝❡ss❛r② s✐♥❝❡ ✇❤❛t ✇❡ ♥❡❡❞ ✐s ♦♥•② t❤❛t S x ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ x✱ ❛♥❞ t❤❛t t❤❡ ♠✐♥✐♠✉♠ ♦❢ p ♦♥ ❛♥② ❜♦✉♥❞❡❞ s❡t ✐s > 0✳ ✾ ❆♣♣•✐❝❛t✐♦♥ t♦ ❢✉♥❝t✐♦♥❛• ❛✉t♦r❡❣r❡ss✐✈❡ ♣r♦❝❡ss❡s ✇✐t❤ ▼❛r❦♦✈ s✇✐t❝❤✐♥❣✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ ♠✐①❡❞ ▼❛r❦♦✈ ♣r♦❝❡ss (I n , X n ) ∈ S ✇❤❡r❡ S = {1, . . . s} × R d ✿ P (I n+1 = j|I n = i) = p ij , 1 ≤ i, j ≤ s ✭✺✽✮ X n+1 = α(I n )ϕ(X n ) + ψ(I n , V n ) ✭✺✾✮ ✇❤❡r❡ α ✐s ❛ ♠❛tr✐① ✈❛•✉❡❞ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥✱ ϕ ❛♥❞ ψ ❛r❡ ✈❡❝t♦r ✈❛•✉❡❞ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥s✱ ❛♥❞ V n ✐s ❛♥ ✐♥❞❡♣❡♥❞❡♥t ✐✳✐✳❞✳ s❡q✉❡♥❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s T f (i, x) = k p ik E[f (k, α(i)ϕ(x) + ψ(i, V 1 ))]. ■❢ ❢♦r ❛•• i t❤❡ ✈❛r✐❛❜•❡ ψ(i, V 1 ) ❤❛s ❛ ❞❡♥s✐t②✱ ✇❡ ❝❛♥ ❛♣♣•② ❈♦r♦••❛r② ✺ ❛t t❤❡ ♣r✐❝❡ ♦❢ ❡①tr❛ r❡❛s♦♥❛❜•❡ ❛ss✉♠♣t✐♦♥s ❜❡❝❛✉s❡ ✭✺✸✮ ❛♥❞ ✭✺✹✮ ✇♦✉•❞ ❜❡ s❛t✐s✜❡❞ ❢♦r s♦♠❡ ❦❡r♥❡• K x ✐♥❞❡♣❡♥❞❡♥t ♦❢ y ✭t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ϕ ✐s ✐♠♣♦rt❛♥t ❤❡r❡✮❀ ♦✉r ♣♦✐♥t ✐s t♦ ❞❡❛• ✇✐t❤ s✐♥❣✉•❛r ♠❡❛s✉r❡s✳ ❆s ✐♥ ❬✷(✱ ❚❤✳✶✳✹✱ ✇❡ ❤❛✈❡ ♠❛❞❡ ❡✛♦rts t♦ ❤❛✈❡ ❝♦♥❞✐t✐♦♥s ✇❤✐❝❤ ❛••♦✇ ❢♦r ♥♦♥✲❝♦♥tr❛❝t✐♥❣ ✈❛•✉❡s ❢♦r α✱ ❛s ❝❛♥ ♦♥❡ s❡❡ ✐♥ ❊q✉❛t✐♦♥ ✭✻✶✮✿ ❚❤❡♦r❡♠ ✻✳ ❈♦♥s✐❞❡r t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ❞❡✜♥❡❞ ❜② ✭✺✽✮ ❛♥❞ ✭✺✾✮✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝❤❛✐♥ I n ✐s ✐rr❡❞✉❝✐❜•❡ ❛♥❞ ❛♣❡r✐♦❞✐❝ ✇✐t❤ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ π ♦♥ ✐ts ✜♥✐t❡ st❛t❡ s♣❛❝❡✱ ❛♥❞ t❤❛t ❢♦r s♦♠❡ q > 0 |ϕ(y) -ϕ(z)| ≤ |y -z| ✭✻✵✮ i π i log( α(i) ) < 0 ✭✻✶✮ sup i E[|ψ(i, V 1 )| q ] < +∞ ✭✻✷✮ ✇❤❡r❡ . ✐s t❤❡ ✉s✉❛• ♠❛tr✐① ♥♦r♠ ❛♥❞ |.| t❤❡ ❡✉❝•✐❞❡❛♥ ♥♦r♠✳ ❚❤❡♥ ❚❤❡♦r❡♠ ✹ ❛♣♣•✐❡s ❛♥❞ ✭✺✵✮ ❤♦•❞s ✇✐t❤ t❤❡ ♥♦r♠ f ′ = sup i,x,x ′ |f (i, x) -f (i, x ′ )| |x -x ′ | η + sup i,x |f (i, x)| |x| η + 1 ❢♦r η s♠❛•• ❡♥♦✉❣❤✳ ❚❤✐s ✐♠♣•✐❡s t❤❛t ❢♦r ❛♥② r❡❛•✐③❛t✐♦♥ (I n , X n ) ♦❢ t❤❡ ❝❤❛✐♥ ❛t t✐♠❡ n ✇✐t❤ ❛♥ ❛r❜✐tr❛r② ✐♥✐t✐❛• ❞✐str✐❜✉t✐♦♥✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ❝♦✉♣•✐♥❣ ✇✐t❤ ❛ ♣❛✐r (I ′ , X ′ ) ❤❛✈✐♥❣ t❤❡ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥✱ s✉❝❤ t❤❛t P (I n = I ′ ) + E[|X n -X ′ | η ] < Cρ n (1 + E[|X 0 | η ]). Pr♦♦❢✳ ❲❡ ✇✐•• ❝❤♦♦s❡ [f ] = i ν i [f ] i , [f ] i = sup x,y |f (i, x) -f (i, y)| |x -y| η v(i, x) = |x| ε e ελ(i) + 1 d((i, x), (j, y)) = 1 i =j + |x -y| η . ❢♦r s♦♠❡ ❝♦♥st❛♥ts ν i ❛♥❞ λ(i) ✇❤✐❝❤ ✇✐•• ❜❡ s♣❡❝✐✜❡❞ •❛t❡r✳ ❈♦♥❝❡r♥✐♥❣ K ❛♥❞ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞s S i,x ✇❡ s✐♠♣•② s❡t✿ K = T S i,x = {i} × R d ■♥ t❤❛t ❝❛s❡ ✭✹✺✮ ❛♥❞ ✭✹✻✮ ❛r❡ ♦❜✈✐♦✉s ✭ε d = 1✮✱ ❛♥❞ ✭✹✼✮ ✇✐•• ❜❡ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭✹✸✮✳ ❚❤❡ t❡❝❤♥✐❝❛• ♣❛rt ✐s t♦ ♣r♦✈❡ t❤❛t ✭✹✸✮ ❛♥❞ ✭✹✹✮ ❤♦•❞ tr✉❡✳ ❲❡ ♥♦✇ ❢♦❝✉s ♦♥ ✭✹✹✮✳ ❲❡ ✜rst ♥♦t❡ t❤❛t s✐♥❝❡ X n+1 = α(I n )(ϕ(X n ) -ϕ(0)) + α(I n )ϕ(0) + ψ(I n , V n ) ✶✵ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ϕ(0) = 0✳ ❯♥s✉r♣r✐s✐♥❣•②✱ t❤❡ ❝♦♥tr❛❝t✐♦♥ ♣r♦♣❡rt② ✭✹✹✮ ✐s r❡•❛t❡❞ t♦ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ t❤❡ ♣r♦❞✉❝t ♦❢ α(I k )✬s ❝♦♥✈❡r❣❡s t♦ ③❡r♦✱ t❤✐s ♦♥❡ ❜❡✐♥❣ ✐ts❡•❢ ❝♦♥tr♦••❡❞ ❜② t❤❡ s♣❡❡❞ ❛t ✇❤✐❝❤ t❤❡ •❛✇ ♦❢ •❛r❣❡ ♥✉♠❜❡rs ❛❝ts ♦♥ t❤❡ s✉♠s ♦❢ log( α(I k ) )✬s✳ ❚❤✐s ✉s❡s ❝•❛ss✐❝❛••② t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✿ ❙✐♥❝❡ t❤❡ ❝❤❛✐♥ I n ✐s ✐rr❡❞✉❝✐❜•❡ ❛♣❡r✐♦❞✐❝ ♦♥ ❛ ✜♥✐t❡ st❛t❡ s♣❛❝❡✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ ❝♦♥st❛♥t✮ s♦•✉t✐♦♥ λ t♦ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ E[λ(I 1 )|I 0 = i] = λ(i) -l(i) + π(l), l(i) = log( α(i) ) ✭✐t ✐s s✐♠♣•② λ = ∞ k=0 (T k 0 -π)l ✇❤❡r❡ T 0 = (p ij ) 1≤i,j≤s ✐s t❤❡ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r ♦❢ t❤❡ ❝❤❛✐♥ I n ✮✳ ❚❤❡ ♣r♦❝❡ss Z n = |X n | ε e ελ(In) s❛t✐s✜❡s✱ t❤❛♥❦s t♦ ✭✺✾✮ ✱✭✻✵✮✱ ❛♥❞ ϕ(0) = 0✿ Z n+1 ≤ ( α(I n ) |ϕ(X n )| + |ψ(I n , V n )|) ε e ελ(In+1) ≤ α(I n ) ε |X n | ε e ελ(In+1) + |ψ(I n , V n )| ε e ελ(In+1) = Z n e ε{log α(In) +λ(In+1)-λ(In)} + e ελ(In+1) |ψ(I n , V n )| ε . ❆♥❞ s✐♥❝❡ t❤❡ ❢❛❝t♦r ♦❢ ε ✐s ❜♦✉♥❞❡❞✱ ✇❡ ❤❛✈❡ ❢♦r s♦♠❡ c Z n+1 ≤ Z n 1 + ε(λ(I n+1 ) -λ(I n ) + log α(I n ) ) + cε 2 + e ελ(In+1) |ψ(I n , V n )| ε E[Z n+1 |F n ] ≤ Z n 1 + επ(l) + cε 2 + e ε sup i λ(i) sup i E[|ψ(i, V 1 )| ε ], ✇❤❡r❡ F n st❛♥❞ ❢♦r t❤❡ σ✲✜❡•❞ σ(I i , X i , 0 ≤ i ≤ n)✳ ❍❡♥❝❡✱ ✐❢ ✇❡ t❛❦❡ ε ≤ q s✉❝❤ t❤❛t επ(l) + cε 2 < 0✱ ✇❡ ♦❜t❛✐♥ ✭✹✹✮✳ ❈♦♥❝❡r♥✐♥❣ ✭✹✸✮✿ |T f (i, y) -T f (i, x)| ≤ k p ik E[|f (k, α(i)ϕ(y) + ψ(i, V 1 )) -f (k, α(i)ϕ(x) + ψ(i, V 1 ))|] ≤ |ϕ(y) -ϕ(x)| η α(i) η k p ik [f ] k [T f ] i ≤ α(i) η k p ik [f ] k i ν i [T f ] i ≤ i,k ν i α(i) η p ik [f ] k . ❲❡ s❡❡ t❤❛t ✐❢ ✇❡ ❝❛♥ ✜♥❞ ν s✉❝❤ t❤❛t ∀ k, i α(i) η ν i p ik < ν k ✭✻✸✮ t❤❡♥ ❊q✉❛t✐♦♥ ✭✹✸✮ ✇✐•• ❜❡ s❛t✐s✜❡❞✳ ❚♦ t❤✐s ❛✐♠✱ ✇❡ ❞❡✜♥❡ ν = π + η k≥1 (π.l -π(l)π)P k ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥ (π.l)(i) = π i l(i). ❙❡t τ i = α(i) η ❀ s✐♥❝❡ τ i = 1 + ηl(i) + O(η 2 ) ✶✶ ✇❡ ❣❡t ν.τ = π + η k≥1 (π.l -π(l)π)P k + ηπ.l + O(η 2 ) ❤❡♥❝❡ ν.τ P = π + η k≥1 (π.l -π(l)π)P k + ηπ(l) + O(η 2 ) = ν + ηπ(l) + O(η 2 ). ❚❤✐s ❡q✉❛t✐♦♥ ✐♠♣•✐❡s t❤❛t ❢♦r η s♠❛•• ❡♥♦✉❣❤✱ ❊q✉❛t✐♦♥ ✭✻✸✮ ✐s s❛t✐s✜❡❞✳ ■♥ ♣❛rt✐❝✉•❛r ✇❡ s❤❛•• ✐♠♣♦s❡ η ≤ ε✳ ❲❡ ❤❛✈❡ ♥♦✇ ♣r♦✈❡❞ ✭✹✸✮ t♦ ✭✹✽✮✳ ❆s ❛ ❜②♣r♦❞✉❝t✱ ❊q✉❛t✐♦♥ ✭✹✸✮ ✐♠♣•✐❡s t❤❛t ❛♥② ❡✐❣❡♥❢✉♥❝t✐♦♥ f ✱ ✇✐t❤ ❛ss♦❝✐❛t❡❞ ❡✐❣❡♥✈❛•✉❡ |λ| = 1✱ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ x✱ ❛♥❞ ❝♦♥s❡q✉❡♥t•②✱ s✐♥❝❡ I n ✐s ✐rr❡❞✉❝✐❜•❡✱ f ✐s ♥❡❝❡ss❛r✐•② ❝♦♥st❛♥t✳ ❚❤❡♦r❡♠ ✹ ❛♣♣•✐❡s ❛♥❞ ✭✺✵✮ ❤♦•❞s ✇✐t❤ f = sup x,i |f (i, x)| |x| ε e ελ(i) + 1 + i ν i [f ] i . ❙✐♥❝❡ ❜② ✐rr❡❞✉❝✐❜✐•✐t②✱ ν i > 0 ❢♦r ❛•• i✱ t❤✐s ♥♦r♠ ✐s ❡q✉✐✈❛•❡♥t t♦ N (f ) = sup i,x |f (i, x)| |x| ε + 1 + sup i,x,y |f (i, x) -f (i, y)| |x -y| η . ❚❤✐s ♥♦r♠ ✐s ❛•s♦ ❡q✉✐✈❛•❡♥t t♦ f ′ ❜❡❝❛✉s❡✱ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞✱ η ≤ ε✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ sup i,x |f (i, x)| |x| η + 1 ≤ sup i,x |f (i, x) -f (i, 0)| + |f (i, 0)| |x| η + 1 ≤ sup i,x |f (i, x) -f (i, 0)| |x| η + sup i |f (i, 0)| ≤ N (f ). ❇② t❤❡ ❞✉❛•✐t② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❲❛ss❡rst❡✐♥ ❞✐st❛♥❝❡ ✭❝❢ ❬✶✼( ❚❤❡♦r❡♠ ✺✳✶✵ ❛♥❞ ❊q✉❛t✐♦♥ ✭✺✳✶✶✮✮ inf In,I ′ ,Xn,X ′ P (I n = I ′ ) + E[|X n -X ′ | η ] = sup f ▲✐♣s❝❤✐t③ E[f (I n , X n ) -f (I ′ , X ′ )] ✇❤❡r❡ t❤❡ ✐♥✜♠✉♠ ✐s t❛❦❡♥ ♦✈❡r ❛•• t❤❡ ♣❛✐rs ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜•❡s (I ′ , X ′ ) ❛♥❞ (I n , X n ) ❤❛✈✐♥❣ r❡s♣❡❝t✐✈❡•② t❤❡ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ❛♥❞ t❤❡ ❝❤❛✐♥ ❞✐str✐❜✉t✐♦♥ ❛t t✐♠❡ n✱ ❛♥❞ f ✐s ✶✲▲✐♣s❝❤✐t③ ✇✳r✳t✳ t❤❡ ❞✐st❛♥❝❡ d✳ ❚❤❡ ❡①♣❡❝t❛t✐♦♥ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s •✉st E[(Q n f )(I 0 , X 0 )]✱ ✇❤✐❝❤ ✐s s♠❛••❡r t❤❛♥ Cρ n (1+E[|X 0 | η ])✳ ✹ ❙✉❜❣❡♦♠❡tr✐❝ r❛t❡s ■♥ t❤❡ r❡st ♦❢ t❤❡ ♣❛♣❡r✱ ✇❡ s❤❛•• ✜♥❞ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ V n t♦ ✵ ✇✐•• ❣✐✈❡ ✉s ❛♥ ✐♥s✐❣❤t ❛❜♦✉t t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ Q n t♦ ✵✳ ❲❡ s❡t ❢♦r ❛♥② ♦♣❡r❛t♦r S ♦♥ E S E0 = sup f ≤1 Sf S 0E = sup f ≤1 Sf . ✶✷ ❲✐t❤ t❤✐s ❝♦♥✈❡♥t✐♦♥✱ ♦♥❡ ❤❛s U V ≤ U E0 V 0E U V ≤ U 0E V E0 . ❲❡ s❤❛•• ❝♦♥s✐❞❡r ♣♦s✐t✐✈❡ r❛t❡ s❡q✉❡♥❝❡s α n ✱ n ≥ 1✱ s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ✭|✶✮ t♦ ✭|✸✮ ❜❡•♦✇✳ ❋♦r ✐♥st❛♥❝❡✱ s❡q✉❡♥❝❡s •✐❦❡ α n = (n + 1) -p ✱ p > 1✱ ♦r α n = exp - √ n✱ ♦r α n = (n + 1) -1 (log(n + 1)) -2 s❛t✐s❢② t❤❡s❡ ❛ss✉♠♣t✐♦♥s ✭♥♦t✐❝❡ t❤❛t t❤❡ ✜rst ♣❛rt ♦❢ ✭|✷✮ ❤♦•❞s ✐❢ x → log α x ✐s ❝♦♥✈❡①✮✳ ❚❤❡s❡ s❡q✉❡♥❝❡s ♠❛❦❡ ✐t ❡❛s② t♦ s♦•✈❡ s♦♠❡ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥s ✭❝❢✳ ❆♣♣❡♥❞✐① ❋✮✳ ❚❤❡♦r❡♠ ✼✳ ▲❡t ✭❆✵✮ ❜❡ s❛t✐s✜❡❞ ❛♥❞ T ❜❡ ❛ . ✲ ❛♥❞ . ✲❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ♦♥ E s❛t✐s❢②✐♥❣ ✭❆✶✮✱ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮✳ ▲❡t α n ❜❡ ❛ s❡q✉❡♥❝❡ s❛t✐s❢②✐♥❣ ✭|✶✮ n → α n ✐s ❞❡❝r❡❛s✐♥❣ ✭|✷✮ n → α n+1 α n ✐s ✐♥❝r❡❛s✐♥❣ ❛♥❞ ❝♦♥✈❡r❣❡s t♦ ✶ ✭|✸✮ n≥1 α 2 n α 2n < ∞. ❲❡ ❛ss✉♠❡ t❤❛t T ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s T = K + V ✇✐t❤ V k E0 ≤ C 1 α k , k > 0, ✭✻✹✮ KV k ≤ C 2 α k , k > 0, ✭✻✺✮ KQ k → 0 ❛s k → ∞ ✭✻✻✮ ✭❊q✉❛t✐♦♥s ✭✻✺✮ ❛♥❞ ✭✻✻✮ ❛r❡ ❝•❡❛r•② s❛t✐s✜❡❞ ✐❢ ✭✻✹✮ ❛♥❞ ✭✷✻✮ ❤♦•❞ tr✉❡✮✳ ❚❤❡♥ ♦♥❡ ❤❛s ❢♦r s♦♠❡ C > 0 ❛♥❞ ❛•• n > 0 Q n E0 ≤ C k≥n α k . ■❢ ✐♥ ❛❞❞✐t✐♦♥ sup n T n < ∞✱ t❤❡♥ Q n E0 ≤ Cα n . ✭✻✼✮ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ ✭✷✼✮ ❛♥❞ ♦♥ t❤❡ ❦❡② r❡s✉•t ♦❢ Pr♦♣♦s✐t✐♦♥ ✶✷✳ ■t ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❊✳ |❡♠❛r❦s✳ ✶✴ ■❢ ❚❤❡♦r❡♠ ✶ ✐s ✉s❡❞ ❢♦r ❝❤❡❝❦✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥s✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❝❤❡❝❦ ✭✷✺✮✱ ✇❤✐❝❤ ✐s ❛✉t♦♠❛t✐❝❛••② s❛t✐s✜❡❞ t❤❛♥❦s t♦ ✭✷✹✮✱ ✭✷✻✮ ❛♥❞ t❤❡ s✉♠♠❛❜✐•✐t② ♦❢ α n ✭❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭|✶✮ ❛♥❞ ✭|✸✮✮✳ ✷✴ ❈♦♥❞✐t✐♦♥ ✭|✷✮ ❡①❝•✉❞❡s ❣❡♦♠❡tr✐❝ r❛t❡s✳ ❚❤❡ t❤❡♦r❡♠ ✐s ✐♥❞❡❡❞ ✇r♦♥❣ ✐♥ t❤✐s ❝❛s❡✿ ❋♦r ❡①❛♠♣•❡ ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛• ❝❛s❡✱ ❚❤❡♦r❡♠ ✶ ❤♦•❞s ✇✐t❤ V = 0✱ ❛♥❞ ✭✻✼✮ ♦♥•② ❤♦•❞s ✇✐t❤ s♦♠❡ ❣❡♦♠❡tr✐❝ r❛t❡✳ ❆♣♣•✐❝❛t✐♦♥ t♦ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❲❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ♦❢ ❚❤❡♦r❡♠ ✷ ❜✉t ✭✸✵✮ ✐s str❡♥❣t❤✲ ❡♥❡❞ ❛s T v(x) ≤ v(x) -θ(v(x)), x / ∈ K 0 ✭✻✽✮ ❢♦r s♦♠❡ ❢✉♥❝t✐♦♥ θ✱ ❡✳❣✳ θ(u) = u q ✱ 0 < q < 1✳ ❖✉r ❣♦❛• ❤❡r❡ ✐s t♦ ✉s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ ❢♦r ❜♦✉♥❞✐♥❣ t❤❡ s❡q✉❡♥❝❡ ρ n ✐♥ ✭✸✽✮✳ ❲❡ ♥❡❡❞ ❛ ♣r❡♣❛r❛t♦r② •❡♠♠❛ ✇❤✐❝❤ ✇✐•• ❜❡ ❡ss❡♥t✐❛• ❢♦r ✇♦r❦✐♥❣ ✇✐t❤ ✭✻✽✮❀ t❤❡ ♣♦✐♥t ♦❢ t❤✐s •❡♠♠❛ ✐s t♦ ❜r✐♥❣ ♦✉t ❛ ❢✉♥❝t✐♦♥ ζ ✇❤✐❝❤ s❛t✐s✜❡s ✭✼✶✮✱ ✐s s✐❣♥✐✜❝❛♥t•② •❛r❣❡r t❤❛♥ ζ(x) = x ❛♥❞ t❤❛t ❝❛♥ ❜❡ ❡❛s✐•② ✐t❡r❛t❡❞ ✭❊q✉❛t✐♦♥ ✭✼✵✮ ✐♠♣•✐❡s ζ (n) (x) = ψ (-1) (ψ(x) + n)✮✿ ✶✸ ▲❡♠♠❛ ✽✳ ▲❡t θ ❜❡ ❛ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♥♦♥✲♥❡❣❛t✐✈❡ ❝♦♥❝❛✈❡ ❞✐✛❡r❡♥t✐❛❜•❡ ❢✉♥❝t✐♦♥ ♦♥ [0, +∞) ✇✐t❤ ❛ ❞❡r✐✈❛✲ t✐✈❡ ✇❤✐❝❤ t❡♥❞s t♦ ③❡r♦ ❛t ✐♥✜♥✐t②✱ ❛♥❞ ❞❡✜♥❡ ❢♦r x ≥ 0 ψ(x) = x 0 1 θ(y) dy ✭✻✾✮ ζ(x) = ψ (-1) (ψ(x) + 1). ✭✼✵✮ ❲❡ ❛ss✉♠❡ t❤❛t ψ ✐s ✜♥✐t❡ ❛♥❞ t❡♥❞s t♦ ✐♥✜♥✐t②✳ ▲❡t θ 1 ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r x ≥ θ 1 θ(x) ≤ x. ❚❤❡♥ ζ ✐s ❝♦♥❝❛✈❡ ❛♥❞ ❢♦r x ≥ θ 1 ζ(x -θ(x)) ≤ x. ✭✼✶✮ ❚❤❡ ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① •✳ ❚❤❡♦r❡♠ ✾✳ ▲❡t ❛•• t❤❡ ❛ss✉♠♣t✐♦♥s ❛♥❞ ♥♦t❛t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✷ ❤♦•❞ ❛♥❞ ❛ss✉♠❡ t❤❛t ✭✸✵✮ ✐s str❡♥❣t❤❡♥❡❞ ❛s T v(x) ≤ v(x) -θ(v(x)), x / ∈ K 0 ✭✼✷✮ ❢♦r s♦♠❡ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ θ s❛t✐s❢②✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ▲❡♠♠❛ ✽ ✇✐t❤ θ 1 = min x (v(x))✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ❛ss✉♠❡ t❤❛t min x≥θ1 x -θ(x) > 0 ❛♥❞ t❤❛t t❤❡ s❡q✉❡♥❝❡ α n = 1 ψ (-1) (n) ✭ψ ✐s ❣✐✈❡♥ ❜② ✭✻✾✮✮ s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥s ✭|✶✮ t♦ ✭|✸✮ ♦❢ ❚❤❡♦r❡♠ ✼✳ ❚❤❡♥ ❢♦r s♦♠❡ c > 0 ❛♥❞ ❛♥② ❜♦✉♥❞❡❞ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ f sup x Q n f (x) v(x) ≤ cρ n f ∞ , ρ n = k≥n α k ✭✼✸✮ π(|Q n f |) ≤ cα n f ∞ . ✭✼✹✮ ❚❤❡ ♣r♦♦❢ ✐s ♣♦st♣♦♥❡❞ t♦ ❆♣♣❡♥❞✐① ❍✳ ❲❡ ✜♥❞ t❤❡ ❢♦••♦✇✐♥❣ ♠❛t❝❤✐♥❣s ❜❡t✇❡❡♥ ❞r✐❢t ❢✉♥❝t✐♦♥ ❛♥❞ r❛t❡s θ(t) α n ρ n log(t + 1) 2 ∼ n -1 (log n) -2 ∼ (log n) -1 t q ✱ 0 < q < 1 ∼ n -1/(1-q) ✱ ∼ n -q/(1-q) ct log(t+1) ∼ e - √ 2cn ∼ e - √ 2cn √ n✳ ❆♣♣•✐❝❛t✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ♦❢ r❡s✉•t ✐♥ t❤❡ ✜❡•❞ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥s ❛r❡ ♥♦t ✉♥❝♦♠♠♦♥✳ ❋♦r ❡①❛♠♣•❡ ✐♥ ❬✶✶( ✭❊①❛♠♣•❡ X n+1 = (X n + W n+1 ) + ✇❤❡r❡ W n ✐s ❛♥ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ✇✐t❤ E[W 1 ] < 0✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ✐♥t❡❣❡r m ≥ 2 s✉❝❤ t❤❛t E[W m 1 ] < ∞ ✶✹ t❤❡② ♣r♦✈❡ t❤❛t t❤❡ ❞r✐❢t ❝♦♥❞✐t✐♦♥ ✭✼✷✮ ✐s s❛t✐s✜❡❞ ✇✐t❤ v(x) = (x + 1) m θ(x) = x α , α = m -1 m . ■♥ ❚❤❡♦r❡♠ ✸✳✻ t❤❡② st❛t❡ t❤❛t ❢♦r ❛♥② x✱ sup f ∞ ≤1 |Q n f (x)| = o(n -α/(1-α) )✱ ✇❤✐❝❤ ✐s s♦♠❡✇❤❛t ✐♥t❡r♠❡✲ ❞✐❛r② ❜❡t✇❡❡♥ ✭✼✸✮ ❛♥❞ ✭✼✹✮✳ ❖♥ t❤✐s ❡①❛♠♣•❡✱ ✇❡ ❝•❡❛r•② s❡❡ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ r❛t❡s ❜❡t✇❡❡♥ ✭✼✸✮ ❛♥❞ ✭✼✹✮✿ ✐❢ t❤❡ ✐♥✐t✐❛• st❛t❡ X 0 = x 0 ✐s ✈❡r② •❛r❣❡✱ ✐t t❛❦❡s ❛ •♦♥❣ t✐♠❡ t♦ ❝♦♠❡ ❜❛❝❦ t♦ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✭t❤✐s t✐♠❡ ✐s ❝❡rt❛✐♥•② ♣r♦♣♦rt✐♦♥❛• t♦ x 0 ✮✱ ❜✉t ✐❢ t❤❡ ✐♥✐t✐❛• st❛t❡ ✐s ❞r❛✇♥ ❢r♦♠ π✱ ✐t ✇♦♥✬t ❜❡ •❛r❣❡ ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✐s ✐♥❝r❡❛s❡❞✳ ❚❤❡② ❛•s♦ ❣✐✈❡ ❛♥ ❛♣♣•✐❝❛t✐♦♥ t♦ ▼♦♥t❡ ❈❛r•♦ ▼❛r❦♦✈ ❈❤❛✐♥s✳ ❲❡ ✇♦✉•❞ •✐❦❡ ♥♦✇ t♦ ❝♦♥✈✐♥❝❡ t❤❡ r❡❛❞❡r t❤❛t ❚❤❡♦r❡♠ ✼ ❝❛♥ ❛•s♦ ❜❡ ❛♣♣•✐❡❞ t♦ ❞②♥❛♠✐❝❛• s②st❡♠s✳ ❆♣♣•✐❝❛t✐♦♥ t♦ ❞②♥❛♠✐❝❛• s②st❡♠s✳ ❈♦♥s✐❞❡r t❤❡ ❢♦••♦✇✐♥❣ ❛♣♣•✐❝❛t✐♦♥ ❞❡✜♥❡❞ ♦♥ [0, 1] v(x) = x(1 + 2 γ x γ ) 0 ≤ x < 1/2 2x -1 1/2 ≤ x ≤ 1 ✭✼✺✮ ✇❤❡r❡ 0 < γ < 1 ✐s ✜①❡❞✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣❡r❛t♦r T f (x) = f (v(x)). ✭✼✻✮ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❛s②♠♣t♦t✐❝s ♦❢ T n ✳ ❚❤❡r❡ ❡①✐sts ❛♥ ❡①t❡♥s✐✈❡ •✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜•❡❝t ❬✶✷✱ ✻( ❛♥❞ t❤❡ r❡s✉•t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ♣r❡s❡♥t ❤❡r❡✱ ❊q✉❛t✐♦♥ ✭✽✵✮✱ ✐s ❛•r❡❛❞② ❦♥♦✇♥ ❬✶✾(❀ ♦✉r ♣♦✐♥t ✐s t♦ ❣✐✈❡ ❛ ♥❡✇ ❛♥❞ ❞✐r❡❝t ♣r♦♦❢ ♦❢ t❤✐s ❡st✐♠❛t❡ ✇❤✐❝❤ ♣•❛②s ❛ ❦❡② r♦•❡ ✐♥ t❤❡ ♦❜t❛✐♥♠❡♥t ♦❢ ❝❡♥tr❛• •✐♠✐t t❤❡♦r❡♠s ✭t❤r♦✉❣❤ t❤❡ •♦r❞✐♥✲▲✐✈❡r❛♥✐ ❚❤❡♦r❡♠✮✱ ❛♥❞ ✇❤✐❝❤ ✐s ❦♥♦✇♥ t♦ ❜❡ ♦♣t✐♠❛• ❬✶✻(✳ ◆♦t✐❝❡ t❤❛t t❤✐s ♣r♦♦❢ ❞♦❡s ♥♦t r❡q✉✐r❡ ❛♥② ❡①♣•✐❝✐t ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✭s❡❡ ❊q✉❛t✐♦♥ ✭✺✳✷✮ ✐♥ ❬✶✷(✮✳ ❲❡ ❞❡t❛✐• ♦♥•② ❤❡r❡ t❤❡ ❡①❛♠♣•❡ ✭✼✺✮ ❜✉t ✐t ✇✐•• ❛♣♣❡❛r ❝•❡❛r•② t❤❛t t❤❡ ❢♦••♦✇✐♥❣ ❞❡✈❡•♦♣♠❡♥t ❡①t❡♥❞s t♦ ♠❛♥② ♦t❤❡r ❝❛s❡s✳ ◆❡✈❡rt❤❡•❡ss✱ ✇❡ ❢❡❡• t❤❛t s✉❝❤ ❡①t❡♥s✐♦♥s ❢❛•• ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ♣❛♣❡r✳ ❋♦r ❛♥② ✐♥t❡❣r❛❜•❡ ❢✉♥❝t✐♦♥ f ♦♥ [0, 1]✱ ✇❡ s❡t F (x) = x 0 f (t)dt -x f , f = 1 0 f (t)dt. ❲❡ st❛rt ✇✐t❤ t❤❡ ❢♦••♦✇✐♥❣ ✐❞❡♥t✐t② ✇❤✐❝❤ ✇❡ ♣r♦✈❡ ❜❡•♦✇✿ T f (x) = v ′ (x) -1 F (v(x)) ′ + f -v ′ (x) -1 ′ F (v(x)) ✭✼✼✮ = V f (x) + Kf (x) ✇❤❡r❡ t❤❡ ♣r✐♠❡ ❞❡♥♦t❡s ✐♥ t❤❡ ✇❤♦•❡ ♣r❡s❡♥t s❡❝t✐♦♥ t❤❡ ❛✳❝✳ ♣❛rt ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥❛• ❞❡r✐✈❛t✐✈❡✳ ❲❡ s❤❛•• t❛❦❡ E = L ∞ ([0, 1])✿ f = f ∞ . ■♥ ♦r❞❡r t♦ ♣r♦✈❡ ✭✼✼✮✱ ♥♦t❡ t❤❛t v ′ (x) -1 F (v(x)) ✐s ❝•❡❛r•② ▲✐♣s❝❤✐t③ ❜❡❝❛✉s❡ F (v(x)) ❝❛♥❝❡•s ❛t t❤❡ ❞✐s❝♦♥✲ t✐♥✉✐t② ♣♦✐♥t ♦❢ v ′ ✱ ✐♠♣•②✐♥❣ t❤❛t t❤✐s ❢✉♥❝t✐♦♥ ❛s ✇❡•• ❛s ✐ts ❞✐str✐❜✉t✐♦♥❛• ❞❡r✐✈❛t✐✈❡ ❜❡•♦♥❣s t♦ E ✇✐t❤ v ′ (x) -1 F (v(x)) ′ = f (v(x)) -f + v ′ (x) -1 ′ F (v(x)) ✇❤✐❝❤ ♣r♦✈❡s ✭✼✼✮✳ ❲❡ ♦❜t❛✐♥ ❛•s♦ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n t❤❛t V n f (x) = v ′ n (x) -1 F (v n (x)) ′ ✭✼✽✮ ✶✺ ✇❤❡r❡ v n ✐s t❤❡ n✲t❤ ✐t❡r❛t❡ ♦❢ v✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤✐s✱ ♥♦t✐❝❡ t❤❛t v ′ n (x) -1 F (v n (x)) ❜❡✐♥❣ ▲✐♣s❝❤✐t③✱ ✐t ✐s t❤❡ ✐♥t❡❣r❛• ♦❢ ✐ts ❞❡r✐✈❛t✐✈❡ ❛♥❞ ✭✼✽✮ •❡❛❞s t♦ V n+1 f (x) = v ′ (x) -1 v ′ n (.) -1 F (v n (.)) (v(x)) ′ = v ′ n+1 (x) -1 F (v n+1 (x)) ′ . ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐t ✐s ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ✐♥ ❛♣♣❡♥❞✐① ■ t❤❛t v ′ n (x) ≥ c 1 n 1/γ v n (x) ✭✼✾✮ ✇✐t❤ c 1 = (2 γ -1) 1/γ ✳ ❍❡♥❝❡✱ ✐❢ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♥♦r♠ f = . 0 f (t)dt ∞ ✱ ✇❡ ❛r❡ •❡❞ t♦ V n f = v ′ n (x) -1 F (v n (x)) ∞ ≤ c -1 1 n -1/γ x -1 F (x) ∞ ≤ c -1 1 n -1/γ sup 0≤x≤1 x -1 x 0 |f (y)|dy ≤ c -1 1 n -1/γ f . ❇❡❝❛✉s❡ B = {f ∈ E : f ≤ 1} ✐s . ✲❝♦♠♣❛❝t ✭F ✐s ✶✲▲✐♣s❝❤✐t③ ✐❢ f ∈ B✮✱ t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✶ ❛♥❞ ♦❢ ❚❤❡♦r❡♠ ✼ ❛r❡ ❛•• s❛t✐s✜❡❞ ✭❜✉t ❤❡r❡ T n ✐s ♥♦t ❜♦✉♥❞❡❞✮✳ ❚❤❛♥❦s t♦ ❝•❛ss✐❝❛• ❞✐st♦rt✐♦♥ ❛r❣✉♠❡♥ts ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❬✶✾( ❚❤❡♦r❡♠ ✶✮✱ ♦♥❡ ❦♥♦✇s t❤❛t T ❛❞♠✐ts ❛ ✉♥✐q✉❡ ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐•✐t② ♠❡❛s✉r❡✱ ✇❤✐❝❤ ✐s ❡r❣♦❞✐❝ ❛♥❞ ♠✐①✐♥❣✳ ■♥ ♣❛rt✐❝✉•❛r✱ t❤❡r❡ ✐s ♥♦ ♥♦♥tr✐✈✐❛• ❡✐❣❡♥❢✉♥❝t✐♦♥ ❢♦r ❛♥② ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s 1 ❛♥❞ ✇❡ ❝❛♥ ❝♦♥❝•✉❞❡ t❤❛t T n f -ν(f ) ≤ Cn 1-1/γ f . ✭✽✵✮ ❆ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ❚❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ r❡q✉✐r❡s t✇♦ ♣r❡•✐♠✐♥❛r② r❡s✉•ts ✇❤✐❝❤ ❛r❡ t❤❡ s✉❜•❡❝t ♦❢ t❤❡ ❢♦rt❤❝♦♠✐♥❣ s❡❝t✐♦♥✳ ❆✳✶ ❆s②♠♣t♦t✐❝❛••② ❛•♠♦st ♣❡r✐♦❞✐❝ ♣♦✇❡rs ♦❢ ❛♥ ♦♣❡r❛t♦r ❚❤❡♦r❡♠ ✶✵ ❜❡•♦✇ ❣✐✈❡s ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤✱ ✐♥ s♦♠❡ s❡♥s❡✱ t❤❡ ♣♦✇❡rs ♦❢ ❛♥ ♦♣❡r❛t♦r T ❝❛♥ ❜❡ r❡✇r✐tt❡♥ T n = i≥1 λ n i P i + T n P 0 ✇❤❡r❡ ❡❛❝❤ P i ✐s ❛ ♣r♦•❡❝t✐♦♥✱ P i P j = 0✱ i = j✱ ❛♥❞ T n P 0 t❡♥❞s t♦ ③❡r♦ ✐♥ s♦♠❡ s❡♥s❡✳ ❍♦✇❡✈❡r✱ ✐❢ ❡❛❝❤ t❡r♠ ♦❢ t❤❡ s❡r✐❡s ✇✐•• ❜❡ ✇❡•• ❞❡✜♥❡❞ ✭❡✐❣❡♥s♣❛❝❡ ❛♥❞ ❡✐❣❡♥✈❛•✉❡✮✱ t❤❡ s❡r✐❡s ♠❛② ❢❛✐• t♦ ❝♦♥✈❡r❣❡✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛•♠♦st ♣❡r✐♦❞✐❝ s❡q✉❡♥❝❡s❀ ❜✉t s✐♥❝❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts x ❢♦r ✇❤✐❝❤ P i x = 0 ❡①❝❡♣t ❢♦r ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✐♥❞✐❝❡s i ✇✐•• ❛♣♣❡❛r t♦ ❜❡ ❞❡♥s❡✱ t❤❡ s❡r✐❡s i≥1 λ n i P i x ✇✐•• ❝♦♥✈❡r❣❡ ❛t •❡❛st ♦♥ ❛ ❞❡♥s❡ s✉❜s♣❛❝❡ ♦❢ E✳ ▲❡♠♠❛ ✶✶ ✇✐•• ❣✐✈❡ ❛ ❝♦♥❞✐t✐♦♥ ✉♥❞❡r ✇❤✐❝❤ t❤❡r❡ ✐s ♦♥•② ❛ ✜♠✐t❡ ♥✉♠❜❡r ♦❢ ♥♦♥ ③❡r♦ λ i ✬s✳ ▲❡t ✉s s❛② ❛ ❢❡✇ ✇♦r❞s ❝♦♥❝❡r♥✐♥❣ ❆ss✉♠♣t✐♦♥s ✭❇✶✮ ❛♥❞ ✭❇✷✮ ❜❡•♦✇✱ s✐♥❝❡ t❤❡② ❛r❡ t❤❡ ❦❡② ❛ss✉♠♣t✐♦♥s ❛♥❞ ♠❛② ❛♣♣❡❛r s♦♠❡❤♦✇ ❝♦♠♣•✐❝❛t❡❞❀ ✐t ✐s ❡❛s✐•② s❤♦✇♥ t❤❛t ✉♥❞❡r t❤❡s❡ ❛ss✉♠♣t✐♦♥s✱ ❢♦r ❛♥② x ∈ E t❤❡ s❡q✉❡♥❝❡ T n x ❤❛s . ✲❝♦♠♣❛❝t ❝•♦s✉r❡✳ ❚❤❡s❡ ❛ss✉♠♣t✐♦♥s ❛r❡ ❡ss❡♥t✐❛••② ✉s❡❞ t♦ ♣r♦✈❡ t❤❡ t♦t❛• ❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡ s❡q✉❡♥❝❡ (T n ) n>0 ❢♦r ❛ ❝❡rt❛✐♥ ♥♦r♠ ✭❙t❡♣ ✶ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✵✮✳ ❚❤❡s❡ ❛ss✉♠♣t✐♦♥s ❛r❡ r❡♠✐♥✐s❝❡♥t ♦❢ t❤❛t ♦❢ t❤❡ ❉❡ ▲❡❡✉✇✲••✐❝❦s❜❡r❣ t❤❡♦r❡♠ ❬✸(✱ ❜✉t ❤❡r❡ ✇❡ ❝♦♥s✐❞❡r . ✲t♦t❛• ❜♦✉♥❞❡❞♥❡ss r❛t❤❡r t❤❛♥ . ✲✇❡❛❦ t♦t❛• ❜♦✉♥❞❡❞♥❡ss ✭✇❤✐❝❤ ✐s ❛❝t✉❛••② ♥♦t ❛ ✇❡❛❦❡r ❛ss✉♠♣t✐♦♥✮✳ ❋♦r t❤❡ st❛t❡♠❡♥t ♦❢ t❤✐s t❤❡♦r❡♠✱ ✇❡ r❡❢❡r t♦ t❤❡ ❡q✉❛t✐♦♥s ✭✶✾✮ t♦ ✭✷✶✮ ❚❤❡♦r❡♠ ✶✵✳ ▲❡t T ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ♦♥ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ (E, . ) s❛t✐s❢②✐♥❣ ❛ss✉♠♣t✐♦♥s ✭❆✵✮✱ ✭❆✶✮ ❛♥❞ ✶✻ ✭❇✶✮ ❚❤❡ s❡q✉❡♥❝❡ T n ✐s ✉♥✐❢♦r♠•② . ✲❡q✉✐❝♦♥t✐♥✉♦✉s ♦♥ . ✲❜♦✉♥❞❡❞ s❡ts ✐♥ t❤❡ ❢♦••♦✇✐♥❣ s❡♥s❡✿ lim x∈B, x →0 sup n T n x = 0 ✭✽✶✮ ✭❇✷✮ T n B ✐s ❛s②♠♣t♦t✐❝❛••② . ✲t♦t❛••② ❜♦✉♥❞❡❞ ✐♥ t❤❡ ❢♦••♦✇✐♥❣ s❡♥s❡✿ ❚❤❡r❡ ❡①✐st ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ s❡ts K n ⊂ E✱ ❛♥❞ ❛ s❡q✉❡♥❝❡ ε n → 0 s✉❝❤ t❤❛t ❢♦r ❛♥② n ≥ 0 T n B ⊂ K n + ε n B 0 . ✭✽✷✮ ❚❤❡♥ t❤❡ ❢♦••♦✇✐♥❣ ❢❛❝ts ❤♦•❞ tr✉❡✿ ❚❤❡ s♣❛❝❡ E ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ . ✲❝•♦s❡❞ s♣❛❝❡s E = {x : T n x → 0} ⊕ {x : lim inf n x -T n x = 0} = E 0 ⊕ E c . ✭✽✸✮ ❚❤❡ ♣r♦•❡❝t✐♦♥ P c ♦♥ E c ♣❛r❛••❡• t♦ E 0 s❛t✐s✜❡s P c ≤ C T ✳ ❚❤❡r❡ ❡①✐st ❛ ♥♦♥✲♥❡❣❛t✐✈❡ s❡q✉❡♥❝❡ ρ n ❝♦♥✈❡r❣✐♥❣ t♦ 0 s✉❝❤ t❤❛t T n x ≤ ρ n x , x ∈ E 0 , n ≥ 0. ✭✽✹✮ ❚❤❡ s♣❛❝❡ E u ♦❢ t❤❡ ✜♥✐t❡ •✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡✐❣❡♥✈❡❝t♦rs ✇✐t❤ ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ♦♥❡ ✐s . ✲❞❡♥s❡ ✐♥ E c ✳ ❚❤❡ s❡t Λ ♦❢ t❤❡s❡ ❡✐❣❡♥✈❛•✉❡s ✐s ❛t ♠♦st ❝♦✉♥t❛❜•❡✱ ❛♥❞ ❢♦r ❡❛❝❤ λ ∈ Λ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ♣r♦•❡❝t✐♦♥ P λ ♦♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥s♣❛❝❡ ♣❛r❛••❡• t♦ t❤❡ ♦t❤❡rs ❛♥❞ t♦ E 0 ✳ ■t s❛t✐s✜❡s P λ ≤ C T ❛♥❞ lim n→∞ P λ x - 1 n n-1 i=0 λ -i T i x = 0, x ∈ E. ✭✽✺✮ ❚❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ k i s✉❝❤ t❤❛t t❤❡ ♣r♦•❡❝t✐♦♥ P c ♦♥ E c s❛t✐s✜❡s lim i→∞ sup x∈B P c x -T ki x = 0. ✭✽✻✮ ❚❤❡ ✉♥✐t ❜❛•• ♦❢ E c ✱ B ∩ E c ✱ ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✳ ■❢ t❤❡ ✐♥t❡❣❡r ♣♦✇❡rs ♦❢ T ❡①t❡♥❞ t♦ ❛ . ✲C 0 ✲s❡♠✐✲❣r♦✉♣ (T t ) t≥0 ✱ ✐✳❡✳ ∀x ∈ E, lim t→0 T t x -x = 0 ✭✽✼✮ t❤❡ s♣❛❝❡ E c ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✈❡❝t♦rs x s✉❝❤ t❤❛t ❢♦r s♦♠❡ ω x ✱ T t x = e iωxt x ❢♦r ❛♥② t ≥ 0✳ Pr♦♦❢✳ ❙t❡♣ ✶✿ ❚❤❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♣♦✇❡rs ♦❢ T ❢♦r♠ ❛ t♦t❛••② ❜♦✉♥❞❡❞ s❡t ❢♦r t❤❡ ❞✐st❛♥❝❡ d(f, g) = sup x ≤1 f (x) -g(x) ♦♥ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ♦♥ B✳ ❆♥② •✐♠✐t ♣♦✐♥t ♦❢ ✐ts ❝•♦s✉r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ♦♥ (E, . )✱ ✇✐t❤ ♥♦r♠ ≤ C T ✳ ❲❡ st❛rt ✇✐t❤ ❛ s✐♠♣•❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ K n ✐♥ ♦r❞❡r t♦ ✐♠❜❡❞ ✐t ✐♥ C T B✳ ❋✐① n > 0✱ ❞❡♥♦t❡ ❜② y k , 1 ≤ k ≤ N n t❤❡ ♣♦✐♥ts ♦❢ K n ✱ ❝❤♦♦s❡ ❛r❜✐tr❛r② N n ♣♦✐♥ts x k ∈ T n B s✉❝❤ t❤❛t x k -y k ≤ ε n ❛♥❞ ❞❡✜♥❡ Kn = {x k , 1 ≤ k ≤ N n }✳ ❆ss✉♠♣t✐♦♥ ✭❇✷✮ ✐s st✐•• s❛t✐s✜❡❞ ✇✐t❤ Kn ❜✉t ε n ✐s ♥♦✇ t✇♦ t✐♠❡s •❛r❣❡r❀ ✐♥ ❛❞❞✐t✐♦♥ Kn ⊂ C T B✳ ❍❡♥❝❡ t❤❡r❡ ❡①✐st t✇♦ ❢✉♥❝t✐♦♥s u n ❛♥❞ v n s✉❝❤ t❤❛t ❢♦r x ≤ 1 T n x = u n (x) + v n (x), u n (x) ∈ Kn , ✶✼ ❛♥❞ v n (x) ≤ 2ε n , v n (x) ≤ 2C T . ✭✽✽✮ ❋✐① n •❛r❣❡❀ ❢♦r ❛♥② p✿ T 2n+p x = C T T n (C -1 T T p u n (x)) + T n+p v n (x) = C T u n (C -1 T T p u n (x)) + C T v n (C -1 T T p u n (x)) + T n+p v n (x) = α p (x) + β p (x) + γ p (x). ❚❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s {α p (.), p ≥ 0} ❤❛s ❛t ♠♦st N Nn n ❡•❡♠❡♥ts❀ ❝•❡❛r•② β p (x) ≤ 2C T ε n ❀ ❛♥❞ ❆ss✉♠♣t✐♦♥ ✭❇✶✮ ✇✐t❤ ❊q✉❛t✐♦♥ ✭✽✽✮ ✐♠♣•✐❡s t❤❛t γ p (x) ≤ η n ✱ ❢♦r ❛•• p ≥ 0 ❛♥❞ s♦♠❡ s❡q✉❡♥❝❡ η n → 0✳ ❲❡ ❤❛✈❡ •✉st ♣r♦✈❡❞ t❤❛t t❤❡ s❡t {T k , k ≥ 2n} ❝❛♥ ❜❡ ❝♦✈❡r❡❞ ✇✐t❤ N Nn n d✲❜❛••s ♦❢ r❛❞✐✉s 2C T ε n + η n ❀ ❤❡♥❝❡ {T k , k ≥ 0} ✐s t♦t❛••② ❜♦✉♥❞❡❞ ❢♦r t❤❡ ❞✐st❛♥❝❡ d✳ ❋♦r ❛♥② x ∈ B✱ t❤❡ s❡q✉❡♥❝❡ T n x ❜❡•♦♥❣s t♦ C T B✱ ❤❡♥❝❡ ❛♥② . ✲❝•✉st❡r ♣♦✐♥t ♦❢ t❤✐s s❡q✉❡♥❝❡ ❜❡•♦♥❣s t♦ C T B ✭❜❡❝❛✉s❡ ♦❢ ✭❆✵✮✮✱ ❛♥❞ t❤❡ ❝♦♥t✐♥✉✐t② ❢♦••♦✇s✳ ❙t❡♣ ✷✿ ❋♦r ❛♥② •✐♠✐ts d(T u k , U ) → 0 ❛♥❞ d(T v k , V ) → 0✱ ♦♥❡ ❤❛s d(T uj +v k , U V ) → 0 ✐❢ min(j, k) → +∞✳ ■♥ ♣❛rt✐❝✉•❛r U V = V U ❛♥❞ ❢♦r ❛♥② t❤✐r❞ s✐♠✐•❛r •✐♠✐t ♦♣❡r❛t♦r W ✱ d(W U, W V ) ≤ C T d(U, V )✳ ❖♥❡ ❤❛s ✐♥❞❡❡❞✿ d(T uj +v k , U V ) ≤ d(T uj +v k , T uj V ) + d(T uj V, U V ) ≤ sup{ T uj x : x ≤ d(T v k , V ), x ≤ 2C T } + d(T uj , U ) V . ❚❤❡ s❡❝♦♥❞ t❡r♠ ♦❜✈✐♦✉s•② ❝♦♥✈❡r❣❡s t♦ ③❡r♦✱ ❛♥❞ t❤❡ ✜rst ♦♥❡ ❛•s♦ ❜❡❝❛✉s❡ ♦❢ ❆ss✉♠♣t✐♦♥ ✭❇✶✮✳ ❋♦r t❤❡ •❛st ❛ss❡rt✐♦♥ d(W U, W V ) = d(U W, V W ) ≤ d(U, V ) W . ❙t❡♣ ✸✿ Pr♦♦❢ ♦❢ ❊q✉❛t✐♦♥s ✭✽✸✮ ❛♥❞ ✭✽✻✮✳ ▲❡t n k ❜❡ ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t T n k d✲❝♦♥✈❡r❣❡s t♦ s♦♠❡ •✐♠✐t S✳ ❲❡ ❝❛♥ ❛ss✉♠❡ t❤❛t n k -n k-1 → ∞✳ ❋r♦♠ t❤❡ s❡q✉❡♥❝❡ n k -n k-1 ♦♥❡ ❝❛♥ ❡①tr❛❝t ❛ s❡q✉❡♥❝❡ p i = n ki+1 -n ki s✉❝❤ t❤❛t T pi ❛♥❞ T pi-1 d✲❝♦♥✈❡r❣❡ t♦ s♦♠❡ •✐♠✐t P c ❛♥❞ R✳ ❙❡t m i = n ki ✳ S = d✲ lim T mi+pi = SP c .
❙✐♥❝❡ p i → ∞✱ t❤❡r❡ ❡①✐sts q i → ∞ s✉❝❤ t❤❛t P c = d✲ lim T mi+qi ❛♥❞ ✇❡ ❣❡t

P c = d✲ lim T mi T qi = d✲ lim ST qi = d✲ lim P c ST qi = P 2 c . P c ✐s ❛ ♣r♦•❡❝t✐♦♥ ♦♥ P c E ❛♥❞ ❊q✉❛t✐♦♥ ✭✽✻✮ ❤♦•❞s✳ ❲❡ s❤❛•• ♣r♦✈❡ ♥♦✇ t❤❛t P c E ✐s ✐♥❞❡❡❞ E c ❛♥❞ t❤❛t ✭✽✸✮ ❤♦•❞s tr✉❡✳ ❈•❡❛r•② P c E ⊂ E c ✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② x ∈ E c t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ r k s✉❝❤ t❤❛t x - T r k x ✳ ❝♦♥✈❡r❣❡s t♦ ✵✳ ❲❡ ❝❛♥ ❛ss✉♠❡ t❤❛t r k > p k ❛♥❞ t❤❛t d(T r k -p k , U ) → 0 ❢♦r s♦♠❡ U ❀ ✐♥ ♣❛rt✐❝✉•❛r d(T r k , P c U ) → 0✳ ❍❡♥❝❡ x = P c U x ∈ P c E✳ ❋✐♥❛••② P c E = E c ✳ ❚❤❡ ♥✉•• s♣❛❝❡ ♦❢ P c ❝•❡❛r•② ❝♦♥t❛✐♥s E 0 ✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❢♦r ❛♥② ♣♦✐♥t x / ∈ E 0 ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ r k s✉❝❤ t❤❛t T r k x ≥ ε ❛♥❞ T r k -p k d✲❝♦♥✈❡r❣❡s t♦ s♦♠❡ •✐♠✐t V ❀ t❤❡ ❜♦✉♥❞ V P c x ≥ ε •❡❛❞s t♦ P c x = 0✳ ❚❤✐s ✐♠♣•✐❡s ❜② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ❛♥② ♣♦✐♥t ♦❢ t❤❡ ♥✉•• s♣❛❝❡ ♦❢ P c ❜❡•♦♥❣s t♦ E 0 ❀ ❤❡♥❝❡ t❤❡ ♥✉•• s♣❛❝❡ ♦❢ P c ✐s E 0 ❛♥❞ E = E 0 ⊕ E c ✳ ❚❤❡ ❜♦✉♥❞ ♦♥ t❤❡ ♥♦r♠ ♦❢ P c ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ •❛st ♣♦✐♥t ♦❢ ❙t❡♣ ✶✳ ❙t❡♣ ✹✿ T ✐s ♦♥❡✲t♦✲♦♥❡ ♦♥ E c ✳ ❚❤❡ ♣♦✇❡rs ♦❢ T ♦♥ E c ❣❡♥❡r❛t❡ ❛ ❝♦♠♣❛❝t G ❣r♦✉♣ ♦❢ ♦♣❡r❛t♦rs ♦♥ E c ✇✐t❤ t❤❡ ❞✐st❛♥❝❡ d c (f, g) = sup x ≤1,x∈Ec f (x) -g(x) .

✶✽ ❙✐♥❝❡

T P c = P c T ❛♥❞ P c = T R = RT ✭R ✐s ❞❡✜♥❡❞ ✐♥ ❙t❡♣ ✸✮✱ E c ✐s T ✲st❛❜•❡ ❛♥❞ R ✐s ✐ts ✐♥✈❡rs❡ ♦♥ E c ✳ ❚❤❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♣♦✇❡rs ♦❢ T G = {T n , n ≥ 0}. ✐s ❛ ❣r♦✉♣ s✐♥❝❡ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t R ∈ G✳ ❚❤❡ ❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♥ G ❝♦♠❡s ❢r♦♠ ❙t❡♣ ✷✱ ❛♥❞ t❤❡ ❝♦♠♣❛❝t♥❡ss ❢r♦♠ ❙t❡♣ ✶✳ ❙t❡♣ ✺✿ E u ✐s . ✲❞❡♥s❡ ✐♥ E c ✳ Pr♦♣❡rt✐❡s ♦❢ P λ ✳ ❊❛❝❤ ❝❤❛r❛❝t❡r χ ♦♥ G ✐s ✉♥✐q✉❡•② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✈❛•✉❡ ♦❢ χ(T )✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ G ❛♥❞ χ(T n ) = χ(T ) n ✳ ❋♦r ❛♥② ❡✐❣❡♥✈❛•✉❡ λ ♦❢ T ✇✐t❤ ♠♦❞✉•✉s ✶✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝❤❛r❛❝t❡r χ s✉❝❤ t❤❛t χ(T ) = λ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❢♦••♦✇s✿ ♣✐❝❦ ❛♥ ❡✐❣❡♥✈❡❝t♦r x✱ ❛ . ✲❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ u s✉❝❤ t❤❛t u(x) = 1 ❛♥❞ s❡t χ(S) = u(Sx)❀ χ ✐s ✐♥❞❡❡❞ ❛ ❝❤❛r❛❝t❡r s✐♥❝❡ ✐t ✐s d c ✲❝♦♥t✐♥✉♦✉s ✇✐t❤ χ(T n ) = χ(T ) n ❀ ✐♥ ♣❛rt✐❝✉•❛r s✐♥❝❡ t❤❡ s❡t ♦❢ ❝❤❛r❛❝t❡rs ♦❢ ❛ ❝♦♠♣❛❝t ❣r♦✉♣ ✐s ❛t ♠♦st ❝♦✉♥t❛❜•❡✱ t❤❡r❡ ✐s ❛t ♠♦st ❛ ❝♦✉♥t❛❜•❡ ♥✉♠❜❡r ♦❢ ❡✐❣❡♥✈❛•✉❡s ♦❢ ♠♦❞✉•✉s ♦♥❡✳ ■♥ ♦r❞❡r t♦ s❤♦✇ ♥♦✇ t❤❛t ❢♦r ❛♥② ❝❤❛r❛❝t❡r χ✱ χ(T ) ✐s ❛♥ ❡✐❣❡♥✈❛•✉❡ ✇❡ ♣r♦❝❡❡❞ ❛s ❢♦••♦✇s✳ ▲❡t µ ❜❡ t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦♥ G✱ ❝♦♥s✐❞❡r ❛ ❝❤❛r❛❝t❡r χ ♦♥ G ❛♥❞ ❞❡✜♥❡ Q χ = G χ(S) -1 Sµ(dS). ✭✽✾✮ ✭❛s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥ G✱ f (S) = S ✐s t❤❡ ✉♥✐❢♦r♠ •✐♠✐t ♦❢ s✐♠♣•❡ ❢✉♥❝t✐♦♥s ✭❜② ❝♦♠♣❛❝t♥❡ss✮ ❛♥❞ t❤✐s ✐♥t❡❣r❛• ✐s ✇❡•• ❞❡✜♥❡❞ ✇✐t❤ t❤❡ ✉s✉❛• ♣r♦♣❡rt✐❡s✱ ❝❢ ❬✺( ■■■✳✷✮✳ ■❢ x ✐s ❛ χ(T )✲❡✐❣❡♥✈❡❝t♦r t❤❡♥ t❤❡ r❡•❛t✐♦♥ T n x = χ(T n )x ❡①t❡♥❞s t♦ G ❛s Sx = χ(S)x✱ ❛♥❞ ❝•❡❛r•② Q χ x = x✳ ❚❤❡ ✐♥✈❛r✐❛♥❝❡ ♦❢ µ ✐♠♣•✐❡s t❤❛t ❢♦r U ∈ G✿ Q χ = G χ(SU ) -1 SU µ(dS) = χ(U ) -1 U Q χ . ✭✾✵✮ ■♥ ♣❛rt✐❝✉•❛r✱ t❛❦✐♥❣ U = T ✱ ❢♦r ❛♥② x ∈ E✱ Q χ x ✐s 0 ♦r ❛♥ ❡✐❣❡♥✈❡❝t♦r ✇✐t❤ ❡✐❣❡♥✈❛•✉❡ χ(T )✳ ■♥ ❛❞❞✐t✐♦♥ ✐♥t❡❣r❛t✐♥❣ t❤✐s ❡①♣r❡ss✐♦♥ ✇✳r✳t✳ µ(dU ) ✇❡ ❣❡t t❤❛t Q χ ✐s ❛ ♣r♦•❡❝t♦r✳ ■❢ Q χ ✐s ♥♦♥✲③❡r♦✱ Q χ ✐s t❤✉s ❛ ♣r♦•❡❝t♦r ♦♥ t❤❡ χ(T )✲❡✐❣❡♥s♣❛❝❡✳ ■❢ Q χ = 0✱ ❢♦r ❛♥② . ✲❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ u ♦♥ E ❛♥❞ y ∈ E✱ ♦♥❡ ❤❛s G χ(S) -1 u(Sy)µ(dS) = 0. ❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ S → u(Sy) ❜❡✐♥❣ ✵✱ t❤✐s d c ✲❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ✐ts❡•❢ ✵✳ ❍❡♥❝❡ u(Sy) = 0 ❢♦r ❛♥② s✉❝❤ u ❛♥❞ y ❛♥❞ ❛♥② S ∈ G✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜•❡✳ ❍❡♥❝❡ Q χ ✐s ♥♦♥✲③❡r♦✱ χ(T ) ✐s ❛♥ ❡✐❣❡♥✈❛•✉❡✱ ❛♥ Q χ ✐s ❛ ♣r♦•❡❝t✐♦♥ ✇❤♦s❡ r❛♥❣❡ ✐s t❤❡ χ(T )✲❡✐❣❡♥s♣❛❝❡✳ ■♥ s✉♠♠❛r②✱ t❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❝❤❛r❛❝t❡rs ❛♥❞ ❡✐❣❡♥✈❛•✉❡s ✇✐t❤ ♠♦❞✉•✉s ♦♥❡✱ ❞❡✜♥❡❞ ❜② λ = χ(T )✱ ❛♥❞ Q χ ✐s ❛ ♣r♦•❡❝t♦r ✇❤♦s❡ r❛♥❣❡ ✐s ❡①❛❝t•② t❤❡ ❡✐❣❡♥s♣❛❝❡✳ ❙✐♥❝❡ S ≤ C T ✇❡ ❤❛✈❡ Q χ ≤ C T ✱ ❛♥❞ s✐♥❝❡ ❜② ✭✽✾✮ t❤❡② ❝♦♠♠✉t❡✱ Q χ ✐s ❛ ♣r♦•❡❝t♦r ♣❛r❛••❡• t♦ t❤❡ ♦t❤❡r ❡✐❣❡♥s♣❛❝❡s✳ ■♥ ♦r❞❡r t♦ s❤♦✇ t❤❛t E u ✐s . ✲❞❡♥s❡ ✐♥ E c ✱ ❝♦♥s✐❞❡r ❛ . ✲❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢♦r♠ u s✉❝❤ t❤❛t u(x) = 0 ❢♦r ❛♥② ❡✐❣❡♥✈❡❝t♦r x✱ t❤❡♥ ❢♦r ❛♥② y ∈ E c ✱ S → u(Sy) ✐s d c ✲❝♦♥t✐♥✉♦✉s ❛♥❞ ❢♦r ❛♥② ❝❤❛r❛❝t❡r χ ♦♥❡ ❤❛s G χ(S) -1 u(Sy)µ(dS) = u(Q χ (y)) = 0. ❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ S → u(Sy) ❜❡✐♥❣ ✵✱ t❤✐s ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐s ✐ts❡•❢ ✵✳ ❍❡♥❝❡ u(y) = 0✳ E u ✐s . ✲❞❡♥s❡ ✐♥ E c ✳ ❚❤❡ ♣r♦•❡❝t✐♦♥ P λ ✐s ✜♥❛••② ✇❡•• ❞❡✜♥❡❞ ♦♥ E ❜② s❡tt✐♥❣ P λ x = Q χ(T ) x ✐❢ x ∈ E c ❛♥❞ P λ x = 0 ✐❢ x ∈ E 0 ✳ ✶✾ ❲❡ ♥♦✇ ♣r♦✈❡ ✭✽✺✮✳ ❚❤✐s ❡q✉❛t✐♦♥ ❤♦•❞s ♦♥ E 0 ❛♥❞ E u ✳ ❙❡t P λ,n = 1 n n-1 i=0 λ -i T i . ❋♦r ❛♥② x ∈ E c ✇❡ ❝❛♥ ♣✐❝❦ ♦✉t y ∈ E u s✉❝❤ t❤❛t x -y ≤ ε ❛♥❞ ❣❡t P λ,q
x -P λ,n x ≤ P λ,q (xy) + P λ,n (xy) + P λ,q y -P λ,n y

≤ 2 sup k T k (x -y) + P λ,q y -P λ,n y . ❙✐♥❝❡ t❤✐s q✉❛♥t✐t② ❝❛♥ ❜❡ ♠❛❞❡ s♠❛••❡r t❤❛♥ 3ε ❜② t❛❦✐♥❣ n ❛♥❞ q •❛r❣❡✱ t❤✐s ♣r♦✈❡s t❤❛t P λ,q x ✐s ❛ . ✲❈❛✉❝❤② s❡q✉❡♥❝❡✱ ❛♥❞ ✐ts •✐♠✐t P λ x s❛t✐s✜❡s ✭✽✺✮✳ ❙✐♥❝❡ ❢♦r ❛•• x ∈ E✱ P λ,n x ≤ C T x ❛♥❞ P λ,n x -P λ x → 0✱ ❆ss✉♠♣t✐♦♥ ✭❆✵✮ ✐♠♣•✐❡s t❤❛t P λ x ≤ C T x ✳ ❙t❡♣ ✻✿ ❊q✉❛t✐♦♥ ✭✽✹✮✳ ❯s✐♥❣ ❛ s❡q✉❡♥❝❡ p k s✉❝❤ t❤❛t d(T p k , P c ) = α k → 0✱ ✇❡ ♦❜t❛✐♥ T p k x ≤ α k ❢♦r x ∈ B ∩ E 0 ✳ ❋♦r n ≥ p k •❛r❣❡✱ ♦♥❡ ❝❛♥ ✇r✐t❡ T n x ≤ T p k (T n-p k x) ≤ C T α k ✳ ❚❤✐s ✐♠♣•✐❡s ✭✽✹✮✳ ❙t❡♣ ✼✿ B c = E c ∩ B ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✳ ❯s✐♥❣ t❤❡ s❛♠❡ s❡q✉❡♥❝❡ p k ✇❡ ❣❡t ✇✐t❤ ✭✽✷✮ B c ⊂ (P c -T p k )B c + T p k B c ⊂ α k B 0 + K p k + ε p k B 0 . ❚❤✐s ♠❡❛♥s t❤❛t B c ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✳ ❙t❡♣ ✽✿ ❈❛s❡ ♦❢ s❡♠✐✲❣r♦✉♣ T t ✳ ❲❡ ❝❛♥ ❝❛rr② ♦♥ ❙t❡♣s ✶ t♦ ✹ ✇✐t❤ t ∈ R + ✐♥st❡❛❞ ♦❢ n ∈ N✳ ❚❤❡ ❣r♦✉♣ G ✐s ♥♦✇ G = {T s , s ≥ 0}✳ ■♥ ❡q✉❛t✐♦♥ (90) ✇❡ t❛❦❡ U = T t ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❛t y = P χ x ✐s ❛ ✈❡❝t♦r s✉❝❤ t❤❛t T t y = χ(T t )y. ■♥ ♣❛rt✐❝✉•❛r ✐❢ y = 0 ✇❡ ❤❛✈❡ χ(T s+t ) = χ(T s )χ(T t )✱ ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❛ss✉♠♣t✐♦♥ ✭✽✼✮ ✐♠♣•✐❡s t❤❛t t❤❡ ❢✉♥❝t✐♦♥ t → T t y ✐s ❝♦♥t✐♥✉♦✉s✱ ❛♥❞ s♦ ✐s t → χ(T t )❀ ❤❡♥❝❡ χ(T t ) = e iωt ❢♦r s♦♠❡ ω ∈ R✳ ❚❤❡ ❢♦••♦✇✐♥❣ •❡♠♠❛ ❣✐✈❡s ❛ ❝♦♥❞✐t✐♦♥ ❢♦r ❝❤❡❝❦✐♥❣ t❤❛t E c ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛•✳ ❚❤✐s ❝♦✉•❞ ❜❡ ❝❤❡❝❦❡❞ s♣❡❝✐✜❝❛••② ♦♥ ❡①❛♠♣•❡s ❜✉t ✇❡ s❤❛•• s❡❡ ✐♥ ❚❤❡♦r❡♠ ✶ t❤❛t t❤✐s ❤♦•❞s ♥❛t✉r❛••② ✐♥ ❣❡♥❡r❛• s✐t✉❛t✐♦♥s❀ ✐♥ ❛❞❞✐t✐♦♥✱ t❤✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛•✐t② ❛ss✉♠♣t✐♦♥ ✐s ✈❡r② ✐♠♣♦rt❛♥t ✐♥ ❚❤❡♦r❡♠ ✼✳ ▲❡♠♠❛ ✶✶✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ t♦ ✭❆✵✮ ❛♥❞ ✭❆✶✮✱ T ✐s . ✲❝♦♥t✐♥✉♦✉s ❛♥❞ s❛t✐s✜❡s t❤❡ ❢♦••♦✇✐♥❣ ❛ss✉♠♣t✐♦♥ ✭❇✶✬✮ ❚❤❡r❡ ❡①✐sts t✇♦ s❡q✉❡♥❝❡s η n → 0 ❛♥❞ η ′ n,p → 0 ✭❛s min(n, p) → ∞✮✱ s✉❝❤ t❤❛t ❢♦r ❛♥② n, p > 0 T n (B ∩ p -1 B 0 ) ⊂ η n B 0 + η ′ n,p B, t❤❡♥ ✭❇✶✮ ✐s ❛•s♦ s❛t✐s✜❡❞✳ ■❢ ✭❇✷✮ ✐s ❛•s♦ s❛t✐s✜❡❞✱ t❤❡♥ ✭✼✮ t♦ ✭✶✶✮ ❤♦•❞ tr✉❡ ❛♥❞ Q n x ≤ ρ n x , ρ n → 0. ✭✾✶✮ Pr♦♦❢✳ ❲❡ st❛rt ✇✐t❤ ✭❇✶✮✳ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ❛♥② s❡q✉❡♥❝❡ x p ♦❢ B s✉❝❤ t❤❛t x p → 0 s❛t✐s✜❡s sup n>0 T n x p → 0✳ ❲✐t❤♦✉t •♦ss ♦❢ ❣❡♥❡r❛•✐t②✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t x p ≤ 1/p✳ ❖♥❡ ❤❛s T n x p ≤ η n + η ′ n,p C 0 . ❙✐♥❝❡ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ T n x p ≤ T n x p ✷✵ ✇❡ ❤❛✈❡ ❢♦r ❛♥② n 0 sup n>0 T n x p ≤ max n≥n0 (η n + η ′ n,p C 0 ) + 1 p max n<n0 T n ✇❤✐❝❤ ❝❛♥ ❜❡ ♠❛❞❡ ❛r❜✐tr❛r✐•② s♠❛•• ❜② t❛❦✐♥❣ n 0 •❛r❣❡ ✜rst ❛♥❞ t❤❡♥ ❜② ✐♥❝r❡❛s✐♥❣ p✳ ▲❡t ✉s ♣r♦✈❡ ♥♦✇ t❤❛t E c ✐s ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛•✳ ■t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t B c = E c ∩ B ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞❀ s✐♥❝❡ ✇❡ ❛•r❡❛❞② ❦♥♦✇ t❤❛t E c ∩ B ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✱ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t . ❛♥❞ . ✐♥❞✉❝❡ t❤❡ s❛♠❡ t♦♣♦•♦❣② ♦♥ B c ✳ ◆♦t✐❝❡ ✜rst t❤❛t ✐❢ x ∈ E ❛♥❞ x -x n → 0 t❤❡♥ x ≤ lim n x n ❜❡❝❛✉s❡ ♦❢ ✭❆✵✮ ✭t❤❡ ✐♥❡q✉❛•✐t② ✐s ♦❜✈✐♦✉s•② tr✉❡ ✐❢ x n ✐s ♥♦t ❜♦✉♥❞❡❞✮✳ ▲❡t x ∈ B ∩ E c ✳ ❲❡ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t x ❝❛♥ ❜❡ ♠❛❞❡ ❛r❜✐tr❛r✐•② s♠❛•• ❜② t❛❦✐♥❣ x s♠❛•• ❡♥♦✉❣❤✳ ❈♦♥s✐❞❡r ❛♥ ✐♥t❡❣❡r p s✉❝❤ t❤❛t x ≤ p -1 ✳ ❚❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ n k s✉❝❤ t❤❛t x -T n k x t❡♥❞s t♦ ③❡r♦✳ ❚❤❛♥❦s t♦ ✭❇✶✬✮✱ t❤❡r❡ ❡①✐st u k ∈ B 0 ❛♥❞ v k ∈ B s✉❝❤ t❤❛t T n k x = η n k u k + η ′ n k ,p v k . ❙✐♥❝❡ x -T n k x + η n k u k t❡♥❞s t♦ ③❡r♦✱ ✉s✐♥❣ t❤❡ ♣r❡✈✐♦✉s r❡♠❛r❦✿ x ≤ lim k T n k x -η n k u k = lim k η ′ n k ,p v k ≤ lim k η ′ n k ,p ✇❤✐❝❤ ❝❛♥ ❜❡ ♠❛❞❡ ❛r❜✐tr❛r✐•② s♠❛•• ❜② t❛❦✐♥❣ p •❛r❣❡✳ ❍❡♥❝❡ . ❛♥❞ . ❛r❡ t♦♣♦•♦❣✐❝❛••② ❡q✉✐✈❛•❡♥t ♦♥ E c ❛♥❞ t❤❡ ❝♦♠♣❛❝t♥❡ss ❤♦•❞s✳ ◆♦✇ t❤❛t E c ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛•✱ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✾✶✮ ❛r❡ ❛♥ ✐♠♠❡❞✐❛t❡ r❡✇♦r❞✐♥❣ ♦❢ t❤❡ ❝♦♥❝•✉s✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✶✵ ✭♥♦t✐❝❡ t❤❛t ρ n ❤❛s ❝❤❛♥❣❡❞ ❢r♦♠ ❡q✉❛t✐♦♥ ✭✽✹✮ ❜② ❛ ❢❛❝t♦r P 0 ✮✳ ❆✳✷ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ▲❡t ✉s r❡❝❛•• t❤❡ ✐❞❡♥t✐t② ✭✷✼✮ T n = n i=1 T n-i (T -V )V i-1 + V n = n i=1 T n-i KV i-1 + V n . ■♥ ♣❛rt✐❝✉•❛r✱ ❆ss✉♠♣t✐♦♥ ✭❆✶✮ t♦❣❡t❤❡r ✇✐t❤ ✭✷✺✮ ✐♠♣•✐❡s t❤❛t t❤❡ s❡q✉❡♥❝❡ V n ✐s ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t C V ✱ ❛♥❞ KV n KB ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞✳ ❲❡ s❡t α n = KV n ❛♥❞ ᾱk = ∞ i=k α i ✳ ▲❡t x ∈ E✱ ❢♦r ❛♥② 0 ≤ k ≤ n✿ (T n -V n )x ≤ n i=1 T n-i KV i-1 x ≤ C T k i=1 KV i-1 x + C T n i=k+1 KV i-1 x ≤ C T C K k i=1 V i-1 x + C T ᾱk x ≤ c k x + C T ᾱk x . ■♥ ♣❛rt✐❝✉•❛r ✐❢ x ∈ B ∩ p -1 B 0 ♦♥❡ ❤❛s (T n -V n )x ≤ min k≤n c k p + C T ᾱk . ✷✶ ❚❤✐s ✐♠♣•✐❡s ✭❇✶✬✮ ✇❤❡r❡ η ′ n,p ✐s t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥ ❛♥❞ η n = ε ′ n ✳ ❲❡ ♣r♦❝❡❡❞ ♥♦✇ ✇✐t❤ ✭✽✷✮✿ T n = n i=1 T i-1 KV n-i + V n = n i=1 ( i-1 j=1 T j-1 KV i-j-1 + V i-1 )KV n-i + V n = 1≤j<i≤n T j-1 KV i-j-1 KV n-i + n i=1 V i-1 KV n-i + V n = A n + B n + C n . ❚❤❡ s❡t A n B ✐s . ✲t♦t❛••② ❜♦✉♥❞❡❞❀ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞ C n B + B n B ⊂ (ε ′ n + n i=1 α n-i ε ′ i-1 )B 0 . ❚❤❡ s✉♠ t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t② ❛♥❞ t❤✐s •❡❛❞s ✜♥❛••② t♦ ✭✽✷✮✳ ❲❡ t✉r♥ ♥♦✇ t♦ t❤❡ •❛st ❛ss❡rt✐♦♥✳ ■❢ T k s❛t✐s✜❡s ✭❇✶✮ ❛♥❞ ✭❇✷✮ ❛♥❞ T ✐s . ✲❝♦♥t✐♥✉♦✉s ❛♥❞ . ✲❝♦♥t✐♥✉♦✉s✱ ❝•❡❛r•② T ❛•s♦ s❛t✐s✜❡s ✭❇✶✮ ❛♥❞ ✭❇✷✮✳ ❚❤❡♦r❡♠ ✶✵ ❛♣♣•✐❡s t♦ T ✳ ❙✐♥❝❡ ❛♥② ❡✐❣❡♥✈❡❝t♦r ♦❢ T ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥ ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ♦♥❡ ✐s ❛♥ ❡✐❣❡♥✈❡❝t♦r ♦❢ T k ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥ ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ♦♥❡✱ E c ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛•✱ ❛♥❞ ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮ ❤♦•❞✳ ❇ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷ ✭❆✵✮ ✐s ❝•❡❛r•② s❛t✐s✜❡❞✳ ■♥ ❛❞❞✐t✐♦♥ T ✐s ❛ . ✲❝♦♥tr❛❝t✐♦♥✱ ❛♥❞ ✭❆✶✮ ❤♦•❞s tr✉❡✳ ❯♣ t♦ ❛ r❡♣•❛❝❡♠❡♥t ♦❢ v ✇✐t❤ v/c v ✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t c v = 1✳ ❙✐♥❝❡ T 1 = 1✱ ❊q✉❛t✐♦♥s ✭✸✵✮✱ ✭✸✶✮ ❛♥❞ ✭✸✷✮ ✐♠♣•② V v ≤ T v ≤ v -1 + c1 K0 ✭✾✷✮ V 1 ≤ 1 -ε1 K0 ✭✾✸✮ ❢♦r s♦♠❡ c > 0✳ ❈♦♠❜✐♥✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s✱ ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ v = v + c/ε s❛t✐s✜❡s V v ≤ v -1. ✭✾✹✮ ▼✉•t✐♣•②✐♥❣ ✭✾✹✮ ❜② V k ❛♥❞ s✉♠♠✐♥❣✱ ✉♣ ✇❡ ♦❜t❛✐♥ V n v + n-1 k=0 V k 1 ≤ v. ✭✾✺✮ ❊q✉❛t✐♦♥ ✭✷✻✮ ✐s ♦❜✈✐♦✉s ❢r♦♠ ✭✸✹✮ ❛♥❞ ✭✸✺✮ ❛♥❞ ❊q✉❛t✐♦♥ ✭✷✺✮ ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭✾✺✮ ❛♥❞ ✭✸✺✮ s✐♥❝❡ KV n = KV n 1 ∞ ≤ ν(V n 1)✳ ❋♦r ✭✷✹✮ ♥♦t✐❝❡ t❤❛t V n 1 ✐s ❛ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s✱ ❜❡❝❛✉s❡ V 1 ≤ 1✱ ❤❡♥❝❡✿ V n 1 ≤ 1 n n-1 k=0 V k 1 ≤ v n . ❛♥❞ ✭✷✹✮ ❤♦•❞s✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ KT p K✳ ◆♦t✐❝❡ ✜rst t❤❛t ✐♥ t❤❡ ❛ss✉♠♣t✐♦♥s ✇❡ ❝❛♥ r❡♣•❛❝❡ ν ✇✐t❤ ν ❞❡✜♥❡❞ ❛s ν(f ) = i≥0 2 -i ν(T i f ), ✷✷ ✇❤✐❝❤ ♠❛❦❡s T ❝♦♥t✐♥✉♦✉s ♦♥ L 1 (ν) ✇✐t❤ ♥♦r♠ ≤ 2✳ ❙❡❝♦♥❞✱ ♥♦t✐❝❡ t❤❛t ν(v) < ∞✱ t❤❛t ✐s ✭✸✺✮ st✐•• ❤♦•❞s✳ ❊q✉❛t✐♦♥ ✭✸✹✮ ✐♠♣•✐❡s t❤❛t t❤❡ ♠❡❛s✉r❡ µ(g) = g(x, y)K(x, dy)ν(dx) ✐s ❛❜s♦•✉t❡•② ❝♦♥t✐♥✉♦✉s ✇✳r✳t✳ ν(dx) ⊗ ν(dy)✱ ❛♥❞ •❡t p(x, y) ❜❡ ✐ts ❞❡♥s✐t②❀ ✐❢ g ❤❛s t❤❡ ❢♦r♠ g(x, y) = h(x)f (y)✱ ♦♥❡ ❤❛s h(x)f (y)p(x, y)ν(dx)ν(dy) = h(x)(Kf )(x)ν(dx) ❤❡♥❝❡ ♦♥❡ ❤❛s ❢♦r ❛♥② ❜♦✉♥❞❡❞ ♠❡❛s✉r❛❜•❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ❢♦r ν✲❛✳❡✳ x✱ Kf (x) = f (y)p(x, y)ν(dy). ❚❤❡ ❢✉♥❝t✐♦♥ p ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✐♥ L 1 (ν ⊗ ν) ❛s p(x, y) = n i=1 q i (x)r i (y) + ρ(x, y), |ρ(x, y)|ν(dx)ν(dy) < ε. ❚❤✐s ✜♥✐t❡ r❛♥❦ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♠♣•✐❡s t❤❛t KB ✐s t♦t❛••② ❜♦✉♥❞❡❞ ✐♥ L 1 (ν)✳ ❇② ❝♦♥t✐♥✉✐t②✱ t❤❡ s❛♠❡ ♣r♦♣❡rt② ❤♦•❞s ❢♦r T p KB✳ ◆❡①t✱ ❊q✉❛t✐♦♥ ✭✸✹✮ ✐♠♣•✐❡s t❤❛t KT p KB ✐s t♦t❛••② ❜♦✉♥❞❡❞ ✐♥ (E, . )✳ ❚❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✶ ❛r❡ t❤✉s s❛t✐s✜❡❞✳ ❚♦ ♦❜t❛✐♥ ✭✸✽✮✱ ✐t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t t❤❡ s♣❛❝❡ E c ✐s ♦♥❡ ❞✐♠❡♥s✐♦♥❛•✳ ❋♦r t❤✐s✱ •❡t n k ❜❡ ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t λ n k i → λ i ❢♦r ❡❛❝❤ ❡✐❣❡♥✈❛•✉❡ λ i ✇✐t❤ ♠♦❞✉•✉s ✶✱ ❛♥❞ ❞❡♥♦t❡ ❜② P c t❤❡ ♣r♦•❡❝t♦r ♦♥ E c ♣❛r❛••❡• t♦ E 0 ✳ ❚❤❡♥ T n k f -T P c f ❝♦♥✈❡r❣❡s t♦ ✵ ❢♦r ❛♥② f ∈ E✳ ❍❡♥❝❡ T P c ✐s ❛ ▼❛r❦♦✈ tr❛♥s✐t✐♦♥ ♦♣❡r❛t♦r ✇✐t❤ t❤❡ s❛♠❡ ♦♥❡✲♠♦❞✉•✉s ❡✐❣❡♥✈❡❝t♦rs ❛s T ✳ ■t ✐s ❝♦♠♣❛❝t ♦♥ E ❛♥❞ ✐❢ t❤❡r❡ ❡①✐sts ♠♦r❡ t❤❛♥ ♦♥❡ ❡✐❣❡♥✈❡❝t♦r✱ ♦♥❡ ❝❛♥ ✜♥❞ t✇♦ ♥♦♥ tr✐✈✐❛• ♠❡❛s✉r❛❜•❡ s❡ts A ❛♥❞ B s✉❝❤ t❤❛t T P c 1 A = 1 B ✭❬✶✺( ❈❤✳✻ ➓ ✸✱ ❚❤✳✸✳✼✮✳ ◆♦t✐❝❡ ♥♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ f = P c 1 A s❛t✐s✜❡s 0 ≤ f ≤ 1 ❛♥❞ ❜② ❏❡♥s❡♥✬s ✐♥❡q✉❛•✐t② T (f n ) ≥ (T f ) n = 1 B . ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ s✐♥❝❡ f n ≤ f ✱ ✇❡ ❤❛✈❡ T (f n ) ≤ T f = 1 B ✱ ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❛t T (f n ) = 1 B ❢♦r ❛•• n > 0❀ •❡tt✐♥❣ n t❡♥❞ t♦ ✐♥✜♥✐t②✱ ✇❡ ❣❡t T (1 f =1 ) = 1 B . ❈ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✸ ❲❡ ❜❡❣✐♥ ✇✐t❤ t❤❡ ❝❛s❡ q = 1✳ ❊•❡♠❡♥t❛r② ✐♥❞✉❝t✐♦♥s •❡❛❞ t♦ T n x ≤ γ n x + c T n-1 x + γc T n-2 x + ... + γ n-1 c x ≤ γ n x + c n x . ✭✾✻✮ ❚❤✐s ♠❛② ❜❡ ✐♠♣r♦✈❡❞ ❛s T n x ≤ C T min k≤n T k x ≤ C T min k≤n γ k x + c k x . ❚❤✐s ✐♠♣•✐❡s ✭❇✶✬✮ ♦❢ ▲❡♠♠❛ ✶✶ ✇✐t❤ η n = 0 ❛♥❞ η ′ n,p = C T min k≤n γ k + c k p . ✭✾✼✮ ❲❡ ❤❛✈❡ s✐♠✐•❛r•② T n B ⊂ γ n B + γ n-1 K B + γ n-2 T K B + • • • + T n-1 K B ✷✸ ❛♥❞ t❤✐s ✐♠♣•✐❡s ♥♦✇ ✭❇✷✮ ✐♥ ❚❤❡♦r❡♠ ✶✵✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t Q ✭❢r♦♠ ❊q✳✭✼✮✮ ❤❛s ❛ s♣❡❝tr❛• r❛❞✐✉s < 1✳ ◆♦t✐❝❡ t❤❛t ❢♦r ❛♥② n > 0✱ Q n = T n-1 Q✱ t❤✐s ♣r♦✈❡s t❤❛t C Q = sup n Q n ✐s ✜♥✐t❡✳ ❋♦r ❛♥② x ∈ B ✇❡ ❤❛✈❡ ❢r♦♠ ✭✼✮✱ ✭✾✻✮ ❛♥❞ ✭✶✽✮ Q n+k x = T n Q k x ≤ (C Q + 1) T n Q k x Q k x + 1 ≤ (C Q + 1)η ′ n,p ✇✐t❤ p -1 = Q k x Q k x + 1 ≤ ρ k . ❇② ❝❤♦♦s✐♥❣ n ❛♥❞ k •❛r❣❡ ❡♥♦✉❣❤✱ t❤✐s ❡♥s✉r❡s t❤❛t s♦♠❡ ♣♦✇❡r ♦❢ Q ✐s ❛ . ✲❝♦♥tr❛❝t✐♦♥✳ ■❢ ♥♦✇ q > 1✱ t❤❡ ♦♣❡r❛t♦r T q s❛t✐s✜❡s t❤❡ ❛ss✉♠♣t✐♦♥s ❢♦r t❤❡ ❝❛s❡ q = 1✱ t❤✉s T q s❛t✐s✜❡s ✭❆✶✮✱ ✭❇✶✬✮ ❛♥❞ ✭❇✷✮✳ ❙✐♥❝❡ T ✐s . ❛♥❞ . ✲❝♦♥t✐♥✉♦✉s✱ t❤✐s ❝•❡❛r•② ✐♠♣•✐❡s t❤❛t T ❛•s♦ s❛t✐s✜❡s t❤❡s❡ ❛ss✉♠♣t✐♦♥s✱ ❜② ✇r✐t✐♥❣ T n = T r+kq ✇✐t❤ 0 ≤ r < q✳ ❉ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✹ ❋♦r t❤❡ ♣r♦♦❢✱ ✇❡ s❤❛•• ❝❤❛♥❣❡ . ✐♥t♦ f ′ = sup x |f (x)| v ′ (x) , v ′ (x) = v(x) + A 1 + A ❢♦r s♦♠❡ ❝♦♥st❛♥t A ≥ 1 ✇❤✐❝❤ ✇✐•• ❜❡ ❝❤♦s❡♥ •❛t❡r✱ ❛♥❞ f ❛s f ′ = f ′ + q[f ] ❢♦r ❛ s♠❛•• ❝♦♥st❛♥t q✱ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✸ ❛r❡ ❢✉•✜••❡❞✳ ◆♦t✐❝❡ t❤❛t f ≤ f ′ ✳ ❋♦r ❛♥② f ∈ E✱ ❜② t❤❡ ♣♦s✐t✐✈✐t② ♦❢ T ❛♥❞ ❊q✉❛t✐♦♥ ✭✹✹✮✱ |T f (x)| ≤ f ′ T v ′ (x) ≤ f ′ γ v v(x) + A + c v 1 + A ≤ f ′ v ′ (x) + c v 1 + A ✭✾✽✮ ❤❡♥❝❡ T f ′ ≤ 1 + c v A f ′ . ✭✾✾✮ T ✐s . ′ ✲❝♦♥t✐♥✉♦✉s✳ ■♥ ❛❞❞✐t✐♦♥✱ ❊q✉❛t✐♦♥ ✭✹✹✮ ✐♠♣•✐❡s t❤❛t ❢♦r ❛♥② n > 0 T n v(x) ≤ γ n v v(x) + c v 1 -γ v ✭✶✵✵✮ ❤❡♥❝❡ T n ′ ✐s ❜♦✉♥❞❡❞✳ ❊q✉❛t✐♦♥ ✭✹✸✮ ✇✐t❤ ✭✾✾✮ ✐♠♣•✐❡s ✭✹✵✮ ✇✐t❤ γ = γ b ✱ ❛♥❞ c = 1 + c d /A✳ ❲✐t❤ ✭✶✵✵✮✱ ✐t ✐♠♣•✐❡s ❛•s♦ t❤❛t T n ′ ✐s ❜♦✉♥❞❡❞✳ ❚❤✉s ✭❆✵✮✱ ✭❆✶✮ ❛♥❞ ✭✹✵✮ ❛r❡ s❛t✐s✜❡❞✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❛t ❚❤❡♦r❡♠ ✸ ❛♣♣•✐❡s✱ ✐t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛t ✭✸✾✮ ❤♦•❞s tr✉❡✳ ❊❛❝❤ S x ❝♦♥t❛✐♥s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉r❤♦♦❞ O x ♦❢ x✳ ❈♦♥s✐❞❡r A 0 > 0 ✇❤✐❝❤ ✇✐•• ❜❡ ❝❤♦s❡♥ •❛r❣❡ ❡♥♦✉❣❤ •❛t❡r❀ ✐❢ v(x) ≤ A 0 t❤❡ s❡t O ′ x = O x ∩ {v < 2A 0 } ✐s st✐•• ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ x ❜❡❝❛✉s❡ v ✐s ❝♦♥t✐♥✉♦✉s✳ ❈♦♥s✐❞❡r ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ (x i ) 1≤i≤I s✉❝❤ t❤❛t v(x i ) ≤ A 0 ❛♥❞ {v ≤ A 0 } ⊂ ∪ I i=1 O ′ xi ✳ ❚❤✐s ✐s ♣♦ss✐❜•❡ t❤❛♥❦s t♦ t❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ {v ≤ A 0 }✳ ❚❤❡r❡ ❡①✐st θ 1 (x), θ 2 (x), . . . , θ I+1 (x) ❛ •♦❝❛••② ▲✐♣s❝❤✐t③ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ✉♥✐t② ♦❢ S s✉❝❤ t❤❛t t❤❡ s✉♣♣♦rt ♦❢ ❡❛❝❤ θ i ✱ i ≤ I✱ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ O ′ xi ✱ ❛♥❞ t❤❡ s✉♣♣♦rt ♦❢ θ I+1 ✐s ❝♦♥t❛✐♥❡❞ ✐♥ {x : v(x) > A 0 } ✭s❡❡ ❬✶( ❚❤✳✷ ♣✳✶✵✮✳ ❲❡ ❞❡✜♥❡ ϕ = 1 -θ I+1 ✇❤✐❝❤ ✐s 0 ♦♥ {v ≥ 2A 0 } ❛♥❞ 1 ♦♥ {v ≤ A 0 }✳ ❲❡ s♣•✐t T f ❛s T f (x) = I i=1 {T f (x) -εϕ(x)K xi f (x i )} θ i (x) + T f (x)θ I+1 (x) + εϕ(x) I i=1 K xi f (x i )θ i (x) = V f (x) + Sf (x) ✷✹ ❈•❡❛r•②✱ ❢♦r f ≤ 1✱ Sf ❜❡•♦♥❣s t♦ ❛ ✜①❡❞ . ✲❝♦♠♣❛❝t s❡t ❜❡❝❛✉s❡ t❤❡ s✉♠ ✐s ✜♥✐t❡✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ s❤♦✇ t❤❛t V f ≤ γ 2 f ✭✶✵✶✮ ❢♦r s♦♠❡ γ 2 < 1❀ t❤✐s ✇✐•• ✐♠♣•② ✭✸✾✮✳ ◆♦t✐❝❡ t❤❛t f → V f ✐s ♥♦t •✐♥❡❛r✱ ❜✉t t❤✐s ❞♦❡s ♥♦t ♠❛tt❡r✳ ❖♥❡ ❤❛s [V f ] ≤ [T f ] + ε i |K xi f (x i )| [ϕθ i ] ≤ γ b [f ] + ε f i (γ v v(x i ) + c v ) [ϕθ i ] ≤ γ b [f ] + εc 0 f ′ , c 0 = (A 0 + c v ) i [ϕθ i ]. ✭✶✵✷✮ ■t ✐s ♠♦r❡ ❝♦♠♣•✐❝❛t❡❞ t♦ ❜♦✉♥❞ V f ′ ✳ ❋♦r i ≤ I ❛♥❞ θ i (x) > 0 t❤❡♥ x ∈ S xi ❛♥❞ ❊q✉❛t✐♦♥ ✭✾✽✮ ❛♥❞ ✭✹✼✮ ✐♠♣•② t❤❛t T f (x) -εϕ(x)K xi f (x i ) = (1 -εϕ(x))T f (x) + εϕ(x)(T f (x) -K xi f (x)) + εϕ(x)(K xi f (x) -K xi f (x i )) ≤ 1 -εϕ(x) γ v v(x) + c v + A f ′ 1 + A + εϕ(x)(T -K xi )v ′ (x) f ′ + εc 1 [f ] ✇❤❡r❡ c 1 ✐s t❤❡ ♠❛①✐♠✉♠ ♦❢ c d ♦♥ {ϕ > 0}✳ ❙✐♥❝❡ ϕ(x) > 0 ✐♠♣•✐❡s v(x) ≤ 2A 0 ✱ ✐❢ ✇❡ ❞❡♥♦t❡ ❜② γ 0 t❤❡ ♠❛①✐♠✉♠ ♦❢ 1 -ε d ♦♥ {v ≤ 2A 0 } t❤❡ s❡❝♦♥❞ t❡r♠ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❛s (T -K xi )v ′ (x) ≤ T v(x) + A -K xi (v + A)(x) 1 + A ≤ γ v v(x) + c v + γ 0 A 1 + A ≤ γ d v ′ (x) ✇✐t❤ γ d = max(γ v , γ 0 + c v /A)✳ ◆♦t✐❝❡ t❤❛t γ d < 1 ❛s s♦♦♥ ❛s A > c v /(1 -γ 0 )✳ ❖✉r ❜♦✉♥❞ ❜❡❝♦♠❡s T f (x) -εϕ(x)K xi f (x i ) ≤ (1 -εϕ(x))(γ v v(x) + c v + A) f ′ 1 + A + εγ d v ′ (x) f ′ + εc 1 [f ] ■❢ ✐♥ t❤✐s ❡①♣r❡ss✐♦♥✱ ϕ(x) < 1✱ t❤❡♥ v(x) ≥ A 0 ❛♥❞ (1 -εϕ(x))(γ v v(x) + c v + A) ≤γ v v(x) + c v + A ≤ sup u≥A0 γ v u + c v + A u + A (v(x) + A) = γ v A 0 + c v + A A 0 + A (v(x) + A) ❛♥❞ ✐❢ ϕ(x) = 1✿ (1 -εϕ(x))(γ v v(x) + c v + A) ≤ (1 -ε) sup u≥0 γ v u + c v + A u + A (v(x) + A) = (1 -ε) c v A + 1 (v(x) + A). ■♥ ❛♥② ❝❛s❡✱ ✇❡ ❣❡t |T f (x) -εϕ(x)K xi f (x i )| ≤ γ 1 f ′ v ′ (x) + εγ d f ′ v ′ (x) + εc 1 [f ] ✭✶✵✸✮ ✇✐t❤ γ 1 = max γ v A 0 + c v + A A 0 + A , (1 -ε) 1 + c v A . ✷✺ ■♥ ♦r❞❡r t♦ ❜♦✉♥❞ t❤❡ ❢❛❝t♦r ♦❢ θ I+1 ✐♥ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ V f ✱ ✇❡ ♥♦t✐❝❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ θ I+1 (x) > 0✱ ✇❡ ❤❛✈❡ v(x) ≥ A 0 ❛♥❞ |T f (x)| ≤ γ v v(x) + c v + A 1 + A f ′ = γ v v ′ (x) + c v + (1 -γ v )A 1 + A f ′ ≤ γ v + c v + (1 -γ v )A A 0 + A f ′ v ′ (x) ≤ c v + A + γ v A 0 A 0 + A f ′ v ′ (x) ≤ γ 1 f ′ v ′ (x). ✭✶✵✹✮ ❙✐♥❝❡ ✭✶✵✸✮ ✐s tr✉❡ ✐❢ θ i (x) > 0✱ ❛♥❞ ✭✶✵✹✮ ❤♦•❞s ✐❢ θ I+1 (x) > 0✱ ✇❡ ♦❜t❛✐♥ ❢♦r ❛•• x |V f (x)| ≤ γ 1 v ′ (x) f ′ + εγ d f ′ v ′ (x) + εc 1 [f ] t❤✉s V f ′ ≤ (γ 1 + εγ d ) f ′ + εc 1 [f ] ✭✶✵✺✮ ❛♥❞ ❝♦♠❜✐♥✐♥❣ ✭✶✵✺✮ ❛♥❞ ✭✶✵✷✮ •❡❛❞s t♦ V f ′ + q[V f ] ≤ γ 1 + εγ d + εqc 0 f ′ + qγ b + εc 1 [f ]. ✭✶✵✻✮ ■♥ ♦r❞❡r t♦ ❣❡t ✭✶✵✶✮ ❢♦r s♦♠❡ γ 2 < 1✱ ✇❡ ♥❡❡❞ s✐♠✉•t❛♥❡♦✉s•②✿ γ v A 0 + c v + A A 0 + A + εγ d + εqc 0 < 1 1 + c v A -ε 1 + c v A -γ d -c 0 q < 1 γ b + ε c 1 q < 1.
■♥ ♦t❤❡r ✇♦r❞s✱ ✐t s✉✣❝❡s t❤❛t

ε γ d + c 0 q < A 0 -γ v A 0 -c v A 0 + A c v < εA 1 -γ d -c 0 q ε c 1 q < 1 -γ b . |❡♠❡♠❜❡r t❤❛t γ d ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ A ❛♥❞ A 0 ✱ ❛❝t✉❛••② 1 -γ d = min 1 -γ v , min v≤2A0 ε d (x) -c v /A . ❋♦r s♦♠❡ •❛r❣❡ ✈❛•✉❡ ♦❢ A 0 ✱ ✇❡ s❡t A = A 2 0 , ε = (1 -γ v )/(2A 0 ), q = (1 -γ d )/2c 0 ✳ ❚❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥s ❜❡❝♦♠❡ (1 -γ v )(1 + γ d ) 4 < A 0 -γ v A 0 -c v 1 + A 0 c v < 1 4 (1 -γ v )A 0 (1 -γ d ) (1 -γ v )c 1 c 0 A 0 (1 -γ d ) < 1 -γ b . ✷✻ ❙✐♥❝❡ ❜② ❛ss✉♠♣t✐♦♥ ✭✹✽✮ ❛♥❞ ✭✹✾✮✱ (1 -γ d )A 0 t❡♥❞s t♦ ✐♥✜♥✐t② ✇✐t❤ A 0 t❤❡s❡ ❡q✉❛t✐♦♥ ❛r❡ s❛t✐s✜❡❞ ❢♦r A 0 •❛r❣❡ ❡♥♦✉❣❤✱ ❛s ✇❡•• ❛s t❤❡ ❝♦♥❞✐t✐♦♥ A ≥ c v /(1 -γ 0 ) t❤❛t ❤❛s ❜❡❡♥ r❡q✉✐r❡❞ ❜❡❢♦r❡✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❡ •❛st ❛ss❡rt✐♦♥✳ ❙✐♥❝❡ ✶ ✐s t❤❡ ♦♥•② ❡✐❣❡♥✈❛•✉❡ ♦❢ ♠♦❞✉•✉s ♦♥❡ ❛♥❞ s✐♥❝❡ ✐ts ♠✉•t✐♣•✐❝✐t② ✐s ♦♥❡✱ t❤❡r❡ ❡①✐sts ❛ •✐♥❡❛r ❢♦r♠ π ♦♥ E s✉❝❤ t❤❛t ✭✺✵✮ ❤♦•❞s✳ ❚❤✐s ❡q✉❛t✐♦♥ ✐♠♣•✐❡s t❤❛t ❢♦r f ∈ E π(f )1 -T n f ≤ Cρ n f ❤❡♥❝❡ |π(f )| 1 ≤ sup k T k f + Cρ n f . ◆♦✇ ✇❡ ❝❛♥ •❡t n t❡♥❞ t♦ ✐♥✜♥✐t② ❛♥❞ ❝♦♥❝•✉❞❡ t❤❛t π ✐s . ✲❝♦♥t✐♥✉♦✉s✳ ❚❤✐s . ✲❝♦♥t✐♥✉♦✉s •✐♥❡❛r ❢✉♥❝t✐♦♥❛• ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ♦❢ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s ❡①t❡♥❞s t♦ ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥❛• ♦♥ C c (S)✱ t❤❡ s❡t ♦❢ ❛•• ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ❢✉♥❝t✐♦♥s ♦♥ S✳ ❇② t❤❡ |✐❡s③ ❚❤❡♦r❡♠✱ t❤❡r❡ ❡①✐sts ❛ ❇♦r❡• ♠❡❛s✉r❡ µ s✉❝❤ t❤❛t π(f ) = µ(f ) ❢♦r ❛♥② f ∈ C c (S)❀ s✐♥❝❡ v ✐s t❤❡ ✐♥❝r❡❛s✐♥❣ •✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ♦❢ C c (S)✱ ✇❡ ❤❛✈❡ π(v) = µ(v) < ∞✳ ❆♥② f ✐♥ E ❜❡✐♥❣ t❤❡ . ✲•✐♠✐t ♦❢ ❝♦♠♣❛❝t•② s✉♣♣♦rt❡❞ ▲✐♣s❝❤✐t③ ❢✉♥❝t✐♦♥s✱ ❜② . ✲❝♦♥t✐♥✉✐t② ♦❢ π ✇❡ ♦❜t❛✐♥ t❤❛t π(f ) = µ(f )✱ f ∈ E✳ ❊ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✼ ▼✉•t✐♣•②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ✭✷✼✮ ❜② P 0 ♦♥ t❤❡ •❡❢t ❛♥❞ ❜② Q q ♦♥ t❤❡ r✐❣❤t ✇❡ ❣❡t Q n+q = n-1 i=1 Q n-i KV i-1 Q q + P 0 KV n-1 Q q + P 0 V n Q q . ✭✶✵✼✮ ❲❡ ❝♦♥s✐❞❡r ✜rst t❤❡ s✐♠♣•❡r ❝❛s❡ ✇❤❡♥ T n ✐s ❜♦✉♥❞❡❞✱ s❛② T n ≤ c✳ ■♥ t❤✐s ❝❛s❡✱ ❝♦♥s✐❞❡r✐♥❣ ❛ s❡q✉❡♥❝❡ n k s✉❝❤ t❤❛t λ n k i ❝♦♥✈❡r❣❡s t♦ ✶✱ ❢♦r i = 1, . . . p ✭t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛ ❝♦♥✈❡r❣✐♥❣ s✉❜s❡q✉❡♥❝❡ λ m k ♦❢ λ m = (λ m i , . . . λ m p ) ❛♥❞ t❛❦✐♥❣ n k = m 2k -m k ✮ ❊q✉❛t✐♦♥ ✭✶✸✮ ✐♠♣•✐❡s t❤❛t ❢♦r ❛♥② x ∈ E p i=1 λ n k i P i x ≤ T n k x + Q n k x ❛♥❞ •❡tt✐♥❣ k t❡♥❞ t♦ ✐♥✜♥✐t②✱ t❤❛♥❦s t♦ ✭✶✽✮✿ p i=1 P i x ≤ c x . ❍❡♥❝❡ P 0 ≤ 1 + c ✐s ✜♥✐t❡✱ ❛♥❞ ❊q✉❛t✐♦♥ ✭✶✵✼✮ •❡❛❞s ❞✐r❡❝t•② t♦ Q n+q E0 ≤ n-1 i=1 Q n-i E0 KV i-1 Q q + P 0 Q q KV n-1 E0 + P 0 Q q V n E0 ❲❡ ♣•❛♥ t♦ ❛♣♣•② Pr♦♣♦s✐t✐♦♥ ✶✷ ♦❢ t❤❡ ❆♣♣❡♥❞✐① ✇✐t❤ u n = Q n E0 ❛♥❞ β i = KV i-1 Q q ❢♦r s♦♠❡ q •❛r❣❡ ❡♥♦✉❣❤✳ ❲❡ r❡♠❛r❦ t❤❛t ✭✶✶✷✮ ✐s s❛t✐s✜❡❞ s✐♥❝❡ KV i Q q = KV i T q P 0 ≤ α i C 2 C T P 0 ✭✶✵✽✮ ❇❡❝❛✉s❡ ♦❢ t❤❡ s✉♠♠❛❜✐•✐t② ♦❢ α i ✭❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭|✶✮ ❛♥❞ ✭|✸✮✮✱ ❛♥❞ ✇✐t❤ t❤❡ ❤❡•♣ ♦❢ t❤❡ ▲❡❜❡s❣✉❡ ❉♦♠✐♥❛t❡❞ ❈♦♥✈❡r❣❡♥❝❡ ❚❤❡♦r❡♠✱ ❊q✉❛t✐♦♥ ✭✶✶✸✮ ✇✐•• ❜❡ s❛t✐s✜❡❞ ❢♦r q •❛r❣❡ ❡♥♦✉❣❤ ✐❢ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❢♦r ❛♥② i ≥ 0 lim q KV i Q q = 0. ✭✶✵✾✮ ✷✼ ❇✉t t❤✐s ✐s ❡❛s✐•② ♦❜t❛✐♥❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ i s✐♥❝❡ ✐t ✐s tr✉❡ ❢♦r i = 0 ❛♥❞ ❢♦r ❛♥② i, q > 0 KV i Q q = KV i-1 (T -K)Q q ≤ KV i-1 Q q+1 + KV i-1 KQ q ≤ KV i-1 Q q+1 + KV i-1 KQ q . ❍❡♥❝❡ Pr♦♣♦s✐t✐♦♥ ✶✷ ❛♣♣•✐❡s ❛♥❞ ✭✻✼✮ ❤♦•❞s✳ ■❢ ♥♦✇ T n ✐s ♥♦t ❜♦✉♥❞❡❞ ✇❡ ❤❛✈❡ t♦ ✇♦r❦ s•✐❣❤t•② ♠♦r❡ ♦♥ ❊q✉❛t✐♦♥ ✭✶✵✼✮✳ ❈♦♥s✐❞❡r f (z) = p i=1 (1 -z λi ). ❙✐♥❝❡ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮ ✐♠♣•② t❤❛t T n = λ n i P i + P 0 Q n ✱ n ≥ 0 ✭t❤✐s ❞✐✛❡rs ❢r♦♠ ✭✶✸✮ ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡ t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❝❛s❡ n = 0✮ ✇❡ ❤❛✈❡ f (T ) = P 0 f (Q)✳ ❍❡♥❝❡ ❛❢t❡r ♠✉•t✐♣•✐❝❛t✐♦♥ ♦♥ t❤❡ •❡❢t ❜② f (Q) ❊q✉❛t✐♦♥ ✭✶✵✼✮ ❜❡❝♦♠❡s f (Q)Q n+q = n-1 i=1 f (Q)Q n-i KV i-1 Q q + f (T )KV n-1 Q q + f (T )V n Q q t❤✉s f (Q)Q n+q E0 ≤ n-1 i=1 f (Q)Q n-i E0 KV i-1 Q q + f (T )KV n-1 Q q E0 + f (T )V n Q q E0 . ✭✶✶✵✮ ❙✐♥❝❡ f (T ) < ∞✱ ✭✶✵✽✮ ✐♠♣•✐❡s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C s✉❝❤ t❤❛t f (T )KV n-1 Q q E0 + f (T )V n Q q E0 ≤ Cα n ❛♥❞ ✇❡ ♦❜t❛✐♥✱ ❛s ❜❡❢♦r❡ ✭❜❡❝❛✉s❡ ✭✶✵✽✮ ❛♥❞ ✭✶✵✾✮ st✐•• ❤♦•❞ tr✉❡✮ t❤❛t f (Q)Q n E0 ≤ C ′ α n . ❙❡t g(z) = 1/f (z) = i≥0 g i z i ✳ ❚❤❡ ♣❛rt✐❛• ❢r❛❝t✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ g ✐♠♣•✐❡s t❤❛t sup i |g i | < ∞✳ ❋♦r ❛♥② n ≥ 0 Q n E0 ≤ Q n g(Q)f (Q) E0 ≤ k Q n+k g k f (Q) E0 ≤ sup i |g i | k Q n+k f (Q) E0 ❤❡♥❝❡ Q n E0 ≤ C k≥n α k . ❋ ❈♦♥✈♦•✉t✐♦♥ ♦❢ s❡q✉❡♥❝❡s Pr♦♣♦s✐t✐♦♥ ✶✷✳ ▲❡t (α n ) n≥1 ❜❡ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ s❛t✐s❢②✐♥❣ ❆ss✉♠♣t✐♦♥s ✭|✶✮ t♦ ✭|✸✮ ♦❢ ❚❤❡♦r❡♠ ✼✱ ❛♥❞ (β i ) i≥1 ❜❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ s❡q✉❡♥❝❡✳ ▲❡t q ❜❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ❛♥❞ (u n ) n≥1 ❜❡ ❛ ♥♦♥✲♥❡❣❛t✐✈❡ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t u n+q ≤ C 0 α n + n-1 i=1 u n-i β i , n ≥ 1 ✭✶✶✶✮ ✷✽ ❢♦r s♦♠❡ C 0 > 0✳ ■❢ sup k β k α k < ∞, ✭✶✶✷✮ ∞ i=1 β i < 1 ✭✶✶✸✮ t❤❡♥ sup n u n α n < ∞. ✭✶✶✹✮ Pr♦♦❢✳ ❙❡t v n = u n α n v * n = sup k≤n v k θ n = α n α n+q C β = sup k β k α k t❤❡♥✱ ❢♦r ❛♥② i 0 ❛♥❞ n > i 0 v n+q ≤ C 0 θ n + θ n n-1 i=1 v n-i α n-i β i α n ≤ C 0 θ n + θ n v * i0 n-1 i=n-i0 α n-i β i α n + θ n v * n n-i0 i=i0 α n-i β i α n + θ n v * n i0 i=1 α n-i β i α n ≤ C 0 θ n + θ n v * i0 C β n-1 i=n-i0 α n-i α i α n + θ n v * n C β n-i0 i=i0 α n-i α i α n + θ n v * n α n-i0 α n i0 i=1 β i ≤ C 0 θ n + θ n v * i0 C β i 0 α 1 α n-i0 α n + θ ′ n v * n C β n-i0 i=i0 α n-i α i α n + i0 i=1 β i ✇❤❡r❡ θ ′ n t❡♥❞s t♦ ✶ ✭❆ss✉♠♣t✐♦♥ ✭|✷✮✮✳ ❇② ❛ss✉♠♣t✐♦♥ ✭|✷✮✱ ❢♦r ❛♥② i✱ t❤❡ s❡q✉❡♥❝❡ j → α j-i /α j ✱ j ≥ i ✐s ❞❡❝r❡❛s✐♥❣✱ ❤❡♥❝❡ ❢♦r i ≤ n/2 ♦♥❡ ❤❛s α n-i α n ≤ α i α 2i t❤✉s ❢♦r 1 ≤ i 0 < n n-i0 i=i0 α n-i α i α n ≤ 2 [n/2] i=i0 α n-i α i α n ≤ 2 [n/2] i=i0 α 2 i α 2i ≤ 2 ∞ i=i0 α 2 i α 2i ❛♥❞ ✇❡ ❣❡t✱ ❢♦r n > i 0 v n+q ≤ C ′ + θ ′ n ρv * n ρ = 2 ∞ i=i0 α 2 i α 2i sup k β k α k + i0 i=1 β i ✷✾ ✇❤❡r❡ C ′ ❞❡♣❡♥❞s ♦♥ ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t ♦♥ n✳ ❙✐♥❝❡ θ ′ n → 1 ❛♥❞ i 0 ❝❛♥ ❜❡ ❝❤♦s❡♥ •❛r❣❡ ❡♥♦✉❣❤ t♦ ❤❛✈❡ ρ < 1✱ t❤✐s ♣r♦✈❡s t❤❛t ❢♦r s♦♠❡ n 0 > 0 ❛♥❞ 0 < ρ ′ < 1 v n+q ≤ C ′ + ρ ′ v * n , n ≥ n 0 ■♥ ♣❛rt✐❝✉•❛r v n+q ≤ C ′ + ρ ′ v * n+q , n ≥ n 0 .
❇② ✐♥❝r❡❛s✐♥❣ C ′ ✇❡ ❡✈❡♥ ❣❡t

v n ≤ C ′′ + ρ ′ v * n , n ≥ 1 ❛♥❞ s✐♥❝❡ t❤❡ r✳❤✳s✳ ✐s ❛•s♦ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r v k ✱ k ≤ n ✭❜❡❝❛✉s❡ v * k ≤ v * n ✮✱ ✇❡ ❣❡t v * n ≤ C ′′ + ρ ′ v * n ,
T (ζ k (v)) ≤ ζ k (T v) ≤ ζ k (v -θ(v) + λ -λV 1) ≤ ζ k (v -θ(v)) + λζ ′ k (v -θ(v))(1 -V 1) ≤ ζ k-1 (v) + λζ ′ k (θ 0 )(1 -V 1), θ 0 = inf x≥θ1 x -θ(x) t❤❡ •❛st ✐♥❡q✉❛•✐t② ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ζ ′ k ✐s ❞❡❝r❡❛s✐♥❣✳ ❚❤✉s✱ s✐♥❝❡ v > θ 0 ✱ V ζ k (v) ≤ T ζ k (v) -ζ k (θ 0 )K1 ≤ ζ k-1 (v) -(ζ k (θ 0 ) -λζ ′ k (θ 0 ))(1 -V 1). ✭✶✶✼✮ ❉✐✛❡r❡♥t✐❛t✐♥❣ ✭✶✶✻✮ ❛♥❞ ✉s✐♥❣ ✭✶✶✺✮✱ ✇❡ ♦❜t❛✐♥ ζ ′ n (x) = ζ ′ (ζ n-1 (x)) ζ ′ n-1 (x) = θ(ζ n (x)) θ(ζ n-1 (x)) ζ ′ n-1 (x)
❤❡♥❝❡

  t✐❛• r❛t❡✱ ❜✉t ✐s s•♦✇❡r✳ ❚❤✐s s✐t✉❛t✐♦♥ ❤❛s ❜❡❡♥ tr❡❛t❡❞ q✉✐t❡ s✉❝❝❡ss❢✉••② ✇✐t❤ ❛ ✈❡r② ♣r♦❜❛❜✐•✐st✐❝ ❛♣♣r♦❛❝❤✱ ✇❤❡r❡ t❤❡ s♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐s r❡•❛t❡❞ t♦ t❤❡ ✐♥t❡❣r❛❜✐•✐t② ♦❢ r❡❝✉rr❡♥❝❡ t✐♠❡s✳ ❚❤❡ r❡❢❡r❡♥❝❡ ❬✶✸(✱ ❛♥❞ ♠♦r❡ s♣❡❝✐✜❝❛••② ❬✶✶(✱ ❞❡❛•s ✇✐t❤ t❤❡s❡ s✐t✉❛t✐♦♥s✳ ❚✇♦ ❦❡② ❝♦♥❝❡♣ts ❛r❡ ✉s❡❞✿ t❤❡ ψ✲✐rr❡❞✉❝✐❜✐•✐t②✱

  ✇❡✈❡r✱ t❤✐s ✐s ♥♦t t❤❡ ❝❛s❡ ❛♥② ♠♦r❡ ✇❤❡♥ t❤❡ ♥♦r♠s ❛r❡ ❞✐✛❡r❡♥t✳ ✶✳✸ ❆✐♠ ♦❢ t❤❡ ♣❛♣❡r ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ s❤♦✇ t❤❛t t❤❡s❡ ✐❞❡❛s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ s✉❝❝❡ss❢✉••② ❛♥❞ t❤❛t t❤❡② •❡❛❞ t♦ ❛♥ ♦♣❡r❛t♦r t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤ ✇❤❡r❡ ♥♦♥ ❣❡♦♠❡tr✐❝ ❝♦♥✈❡r❣❡♥❝❡ ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ♠❛✐♥ ❢❡❛t✉r❡ ♦❢ t❤✐s t❤❡♦r② ✐s t♦ ✇♦r❦ s✐♠✉•t❛♥❡♦✉s•② ✇✐t❤ t✇♦ ♥♦r♠s ❛♥❞ t♦ ✉s❡ t❤✐s ❢♦r ♠❡❛s✉r✐♥❣ ♥♦♥ ❣❡♦♠❡tr✐❝ r❛t❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡✳ ❖✉r ❛♣♣r♦❛❝❤ ❤❛s ❡ss❡♥t✐❛••② t✇♦ st❡♣s✿ ✇❡ ✜rst ❣✐✈❡ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ ✭✼✮ t♦ ✭✶✶✮ ❤♦•❞ ✇✐t❤

ζ❍

  n ≥ 1 ✇❤✐❝❤ ♣r♦✈❡s t❤❛t v n ✐s ❜♦✉♥❞❡❞✳ • Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✽ ❚❤❡ ❡q✉❛t✐♦♥ ψ(ζ(x)) = ψ(x) + 1 ✐♠♣•✐❡s t❤❛t ζ(x) > x✳ ❇② ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤✐s ❡q✉❛t✐♦♥✱ ✇❡ ❣❡t ′′ (x) = θ ′ (ζ(x))ζ ′ (x)θ(x)θ(ζ(x))θ ′ (x) θ(x) 2 = θ(ζ(x)) θ(x) 2 (θ ′ (ζ(x))θ ′ (x)) ≤ 0. ❲❡ t✉r♥ ♥♦✇ ❊q✉❛t✐♦♥ ✭✼✶✮❀ s✐♥❝❡ ψ ✐s str✐❝t•② ✐♥❝r❡❛s✐♥❣✱ ✭✼✶✮ ✐s ❡q✉✐✈❛•❡♥t t♦ ψ(xθ(x)) + 1 ≤ ψ(x) ❜✉t s✐♥❝❡ θ ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ψ(x)ψ(xθ(x)) Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✾ ❲❡ ♣•❛♥ t♦ ❛♣♣•② ❚❤❡♦r❡♠ ✼ ✇✐t❤ f = f ∞ f = f v . ❈•❡❛r•②✱ s✐♥❝❡ ❚❤❡♦r❡♠ ✷ ❛♣♣•✐❡s✱ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮ ❛♥❞ ✭✶✽✮ ❛r❡ s❛t✐s✜❡❞✳ ❆s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷ ✇❡ s❡t V = T -K❀ ✇❡ r❡❝❛•• t❤❛t K(x, S) = 0 ✐❢ x / ∈ K 0 ✭❝❢✳ t❤❡ st❛t❡♠❡♥t ♦❢ ❚❤❡♦r❡♠ ✷✮✳ ❲❡ ❤❛✈❡ t♦ ❡st✐♠❛t❡ V n E0 ✳ |❡❝❛•• t❤❛t ✇❡ ❤❛✈❡ T v ≤ vθ(v) + c1 K0 ✸✵ ❢♦r s♦♠❡ c > 0✱ ❛♥❞ ❊q✉❛t✐♦♥ ✭✸✷✮ ✇✐t❤ t❤❡ ❢❛❝t t❤❛t K(x, S) = 0 ❢♦r x / ∈ K 0 ✐♠♣•② t❤❛t V 1 ≤ 1 -ε1 K0 . ❈♦♠❜✐♥✐♥❣ t❤❡s❡ ❡q✉❛t✐♦♥s ✇❡ ❣❡t T v ≤ vθ(v) + λ(1 -V 1), λ = c ε . ❲❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥s ζ ❛♥❞ ψ ❢r♦♠ θ ❛s ✐♥ ▲❡♠♠❛ ✽ ❛♥❞ ✇❡ s❡t ❢♦r x ≥ 0 ζ n (x) = ψ (-1) (ψ(x) + n) = ζ(ζ n-1 (x)). ✭✶✶✻✮ ❚❤❡ ❢✉♥❝t✐♦♥ ζ n ✐s ❝♦♥❝❛✈❡✱ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ✐♥❝r❡❛s✐♥❣ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥s✳ ❯s✐♥❣ t❤❡ ❏❡♥s❡♥ ✐♥❡q✉❛•✐t② ❛♥❞ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ ζ k ✇❡ ♦❜t❛✐♥

  ✐♥ ❛❞❞✐t✐♦♥ ζ n (θ 0 ) t❡♥❞s t♦ ✐♥✜♥✐t② ❛♥❞ θ(x)/x t❡♥❞s t♦ ③❡r♦ ✭θ ✐s ❝♦♥❝❛✈❡ ✇✐t❤ ❛ ❞❡r✐✈❛t✐✈❡ ✇❤✐❝❤ t❡♥❞s t♦ ③❡r♦✮✱ t❤❡ s❡q✉❡♥❝❡ ζ ′ n (θ 0 )/ζ n (θ 0 ) t❡♥❞s t♦ 0✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡r❡ ❡①✐st n 0 s✉❝❤ t❤❛t λζ ′k (θ 0 )-ζ k (θ 0 ) ≤ 0 ❢♦r k > n 0 ✱ ❤❡♥❝❡ ♠✉•t✐♣•②✐♥❣ ❜♦t❤ s✐❞❡s ♦❢ ✭✶✶✼✮ ❜② V k-1 ❛♥❞ s✉♠♠✐♥❣ ✉♣ ❢r♦♠ ✶ t♦ n > n 0 ✱ ✇❡ ❣❡t V n ζ n (v) ≤ v + c ❢♦r s♦♠❡ ❝♦♥st❛♥t c✳ ❙✐♥❝❡ ζ n (x) ≥ ψ (-1) (n) ✇❡ ❣❡t V n 1 ≤ v + c ψ (-1) (n) . ❚❤❡♦r❡♠ ✼ ❛♣♣•✐❡s ❛♥❞✱ ✐♥ ♣❛rt✐❝✉•❛r✱ ✇❡ ♦❜t❛✐♥ ✭✼✸✮✳ ❋♦r ✭✼✹✮✱ ✇❡ ❝♦♥s✐❞❡r f = π(|f |). ❙✐♥❝❡ . ✐s ✉♥❝❤❛♥❣❡❞✱ ❊q✉❛t✐♦♥s ✭✼✮ t♦ ✭✶✶✮✱ ✭✻✺✮ ❛♥❞ ✭✻✻✮ ❛r❡ st✐•• s❛t✐s✜❡❞✱ ❛s ✇❡•• ❛s ✭✶✽✮ ❜❡❝❛✉s❡ π(|f |) ≤ f v π(v)✳ ■♥ ❛❞❞✐t✐♦♥ T n = 1✱ ❛♥❞ V n E0 = π(V n 1) ≤ c ′ ψ (-1) (n) . ❚❤❡♦r❡♠ ✼ st✐•• ❛♣♣•✐❡s ❛♥❞ ✇❡ ♦❜t❛✐♥ ✭✻✺✮✳ ✸✶ ■ Pr♦♦❢ ♦❢ ❊q✉❛t✐♦♥ ✭✼✾✮ ❲❡ s❤❛•• ♣r♦✈❡ t❤❛t ❢♦r 0 ≤ x < 1 v ′ n (x) γ ≥ 1 + anv n (x) γ , a = 2 γ -1. ✭✶✶✽✮ ❲❡ r❡❝❛•• t❤❛t v(x) = x(1 + 2 γ x γ ) 0 ≤ x < 1/2 2x -1 1/2 ≤ x ≤ 1 ✭✶✶✾✮ ❛♥❞ t❤❛t t❤❡ ♣r✐♠❡ s✐❣♥ st❛♥❞s ❢♦r t❤❡ r✐❣❤t ❞❡r✐✈❛t✐✈❡✳ ■♥ t❤❡ ❝❛s❡ n = 0 t❤❡ ✐♥❡q✉❛•✐t② ✐s ♦❜✈✐♦✉s✳ ■♥ t❤❡ ❝❛s❡ n ≥ 1✱ ✇❡ ❛ss✉♠❡ ❜② ✐♥❞✉❝t✐♦♥ t❤❛t ✭✶✶✽✮ ✐s s❛t✐s✜❡❞ ❛♥❞ s✐♥❝❡ v ′ n+1 (x) = v ′ n (x)v ′ (v n (x))✮✱ ✈❛•✐❞ ❢♦r n ≥ 0✱ ❊q✉❛t✐♦♥ ✭✶✶✽✮ ✇✐t❤ n + 1 ✇✐•• ❜❡ ✐♠♣•✐❡❞ ❜② (1 + anv n (x) γ )v ′ (v n (x)) γ ≥ 1 + a(n + 1)v n+1 (x) γ . ❚❤✐s ❤❛s t♦ ❜❡ ♣r♦✈❡❞ ❢♦r n ≥ 0✳ ■t s✉✣❝❡s t♦ s❤♦✇ t❤❛t ❢♦r ❛♥② 0 ≤ y ≤ 1 (1 + any γ )v ′ (y) γ ≥ 1 + a(n + 1)v(y) γ ✭✶✷✵✮ ✭✐✳❡✳ y = v n (x)✮✳ ❇② •✐♥❡❛r✐t② ♦❢ ❜♦t❤ s✐❞❡s ♦❢ ✭✶✷✵✮ ✇✳r✳t✳ n✱ ✇❡ ♦♥•② ❤❛✈❡ t♦ ❝❤❡❝❦ t❤✐s ❢♦r n = 0✱ ❛♥❞ n → ∞✱ t❤❛t ✐s v ′ (y) γ ≥ 1 + av(y) γ yv ′ (y) ≥ v(y) ✭✶✷✶✮ ✭t❤❡ ✜rst ❡q✉❛t✐♦♥ ✐s ✭✶✶✽✮ ✇✐t❤ n = 1✮✳ ■♥ t❤❡ ❝❛s❡ y < 1/2 t❤✐s ✐s r❡✇r✐tt❡♥ ❛s (1 + (γ + 1)2 γ y γ ) γ ≥ 1 + ay γ (1 + 2 γ y γ ) γ 1 + (γ + 1)2 γ y γ ≥ 1 + 2 γ y γ . ❚❤❡ s❡❝♦♥❞ ✐♥❡q✉❛•✐t② ✐s ♦❜✈✐♦✉s✳ ❋♦r t❤❡ ✜rst ♦♥❡✱ s✐♥❝❡ 2y < 1✱ s❡tt✐♥❣ z = 2 γ y γ ✱ t❤✐s ❤♦•❞s ✐❢ (1 + (γ + 1)z) γ ≥ 1 + az ❢♦r 0 ≤ z ≤ 1✳ ❙✐♥❝❡ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ❜♦t❤ s✐❞❡s ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ z ✇❤✐❝❤ ✈❛♥✐s❤❡s ❛t z = 0✱ ❛♥❞ ✐s ♥♦♥ ♥❡❣❛t✐✈❡ ❛t z = 1 ✭✇❡ r❡❝❛•• t❤❛t a = 2 γ -1✮✱ t❤❡ ✐♥❡q✉❛•✐t② ✐s s❛t✐s✜❡❞✳ ■♥ t❤❡ ❝❛s❡ y ≥ 1/2✱ ✭✶✷✶✮ ✐s 2 γ ≥ 1 + a(2y -1) γ 2y ≥ 2y -1 ✇❤✐❝❤ ✐s ♦❜✈✐♦✉s•② s❛t✐s✜❡❞✳ |❡❢❡r❡♥❝❡s ❬✶( ❏❡❛♥✲P✐❡rr❡ ❆✉❜✐♥ ❛♥❞ ❆rr✐❣♦ ❈❡••✐♥❛✱ ❉✐✛❡r❡♥t✐❛• ✐♥❝•✉s✐♦♥s✱ •r✉♥❞•❡❤r❡♥ ❞❡r ▼❛t❤❡♠❛t✐s❝❤❡♥ ❲✐s✲ s❡♥s❝❤❛❢t❡♥ ❬❋✉♥❞❛♠❡♥t❛• Pr✐♥❝✐♣•❡s ♦❢ ▼❛t❤❡♠❛t✐❝❛• ❙❝✐❡♥❝❡s(✱ ✈♦•✳ ✷✻✹✱ ❙♣r✐♥❣❡r✲❱❡r•❛❣✱ ❙❡t✲✈❛•✉❡❞ ♠❛♣s ❛♥❞ ✈✐❛❜✐•✐t② t❤❡♦r②✳ ❬✷( ❇✳ ❈•♦❡③ ❛♥❞ ▼✳❍❛✐r❡r✱ ❊①♣♦♥❡♥t✐❛• ❡r❣♦❞✐❝✐t② ❢♦r ♠❛r❦♦✈ ♣r♦❝❡ss❡s ✇✐t❤ r❛♥❞♦♠ s✇✐t❝❤✐♥❣✱ ❛r❳✐✈✿✶✸✵✸✳✻✾✾✾✈✶ ✭✷✵✶✸✮✳ ❬✸( ❑✳ ❞❡ ▲❡❡✉✇ ❛♥❞ ■✳ ••✐❝❦s❜❡r❣✱ ❆♣♣•✐❝❛t✐♦♥s ♦❢ ❛•♠♦st ♣❡r✐♦❞✐❝ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥s✱ ❆❝t❛ ▼❛t❤✳ ✶✵✺ ✭✶✾✻✶✮✱ ✻✸✕✾✼✳ ✸✷

❚❤❡♥ ♦♥❡ ❝❛♥ ✇r✐t❡ T f (x) = f (y)ν(dy) + f (y)(≤ 1ν(S)