Emmanuelle Clément

Arnaud Gloter

Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes

Keywords: MSC 2010. Primary: 60G51, 62F12, 60H07. Secondary: 60F05, 60G52, 60J75 Local Asymptotic Mixed Normality Property, Lévy process, stable process, Malliavin calculus for jump processes

We prove the Local Asymptotic Mixed Normality property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process. The process is observed on the fixed time interval [0, 1] and the parameter appears in the drift coefficient only. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof of this result contains a sharp study of the asymptotic behavior, in small time, of the transition probability density of the process and of its logarithm derivative.

Introduction

An important concept in parametric estimation is the Local Asymptotic Mixed Normality property introduced by Jeganathan in a serie of papers ( [START_REF] Jeganathan | On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal[END_REF], [START_REF] Jeganathan | Some asymptotic properties of risk functions when the limit of the experiment is mixed normal[END_REF]), which permits to extend the Le Cam and Hajek's results (see [START_REF] Hájek | A characterization of limiting distributions of regular estimates[END_REF], [START_REF] Le | Asymptotics in statistics[END_REF]) to situations where the local Asymptotic Normality does not hold. Let {E n , E n , (P θ n ) θ∈Θ⊂R d } be a statistical experiment, we say that the LAMN property holds at θ with information matrix I(θ) and rate u n (u n tends to zero as n goes to infinity) if

log dP θ+unh n dP θ n = h T I n (θ) 1/2 N n - 1 2 h T I n (θ)h + o P θ n (1)
where (N n , I n (θ)) converges in law (under P θ n ) to (N, I(θ)) with N a standard gaussian vector independent of I(θ), and I(θ) > 0 a.e. The LAN property is obtained when the information matrix I(θ) is non random.

If the LAMN property is satisfied at θ, then from the Hajek's convolution theorem, we know that for any regular estimator θn such that

u -1 n [ θn -θ] ⇒ Z θ (in law under P θ n ),
Z θ admits the decomposition Z θ = I(θ) -1/2 N + R with N a standard gaussian vector and R independent of N conditionally on I(θ). As a consequence, the minimal asymptotic estimation error is a mixed normal variable with variance I(θ) -1 .

In this paper, we consider the statistical experiment {R n , B n , (P θ n ) θ∈Θ⊂R }, corresponding to the observation of a Lévy driven stochastic equation at discrete times t i = i n , for 1 ≤ i ≤ n. More precisely, we observe (X θ i n ) 1≤i≤n , where (X θ t ) t∈[0,1] is a continuous time process depending on an unknown real parameter θ. There is a large literature concerning the estimation of the parameters, and the LAN property, of a translated Lévy process

X θ t = θ 1 t + θ 2 L t , θ = (θ 1 , θ 2 ),
see for example Aït-Sahalia and Jacod [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF] [2], Masuda [START_REF] Masuda | Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density[END_REF], Kawai and Masuda [START_REF] Kawai | On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling[END_REF], [START_REF] Kawai | Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling[END_REF]. In this case, the statistical study is based on the fact that the density of X θ t can be expressed as a function of the density of L t .

Here, we intend to consider the more general stochastic equation

X θ t = x 0 + t 0 b(X θ s , θ)ds + L t (1)
where (L t ) t∈[0,1] is a pure jump Lévy process, and focus on the estimation of the drift parameter.

When (X θ t ) t is solution of (1), the transition density of X θ t is unknown, and the link between the density of L t and the density of X θ t is not clear. This complicates the statistical study considerably and to our knowledge, there are no results about the asymptotic behavior of the log-likelihood of the discretized process (X θ i/n ) 1≤i≤n .

In this paper, we prove the LAMN property based on the observations (X θ i n

) i where (X θ t ) t∈[0,1] is solution of [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF], with rate u n = n 1/2-1/α , when the Lévy measure of (L t ) is an α-stable Lévy measure near zero, with α ∈ [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF][START_REF] Aït | Fisher's information for discretely sampled Lévy processes[END_REF]. This result is obtained through a representation of the transition density of X θ t , using the Malliavin calculus for jump processes developed by Bichteler, Gravereaux and Jacod [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF]. The recourse to the Malliavin calculus to prove the LAMN property, in a high frequency data setting, has been initiated by Gobet [6] for diffusion processes. However, the situation given by ( 1) is completely different. Indeed, for diffusion processes, it is well known that one can not estimate the drift parameter from the observation of the process on a fixed time interval.

Besides the statistical application, a main contribution of this paper is to precise the asymptotic behavior of the transition density of X θ t , in small time, and of its logarithm derivative with respect to the parameter.

The paper is organized as follows. The main results are stated in Section 2. Section 3 gives some representations of the transition density and its logarithm derivative, using the Malliavin calculus proposed in [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] and Section 4 studies their asymptotic behavior. The proof of the LAMN property is given in Sections 5 and 6. We stress on the fact that contrarily to [START_REF] Gobet | Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach[END_REF], this proof does not require some lower bounds for the density of X θ t . Section 7 contains some more technical proof.

Main results

We consider the real process (X θ t ) defined on the time interval [0, 1], by

X θ t = x 0 + t 0 b(X θ s , θ)ds + L t , (2)
where (L t ) is a centered Lévy process defined on a filtered space (Ω, G, (G t ) t , P ). We assume that the Lévy measure of (L t ) is absolutely continuous with respect to the Lebesgue measure and admits a density F (z) given on R * by We assume that the function b is bounded, with bounded derivatives up to order three with respect to both variables x and θ. Under these assumptions, we know that for t > 0, X θ t admits a density (see Bichteler, Gravereaux and Jacod [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF], Ishikawa and Kunita [START_REF] Ishikawa | Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps[END_REF], Picard [START_REF] Picard | On the existence of smooth densities for jump processes[END_REF], Fournier and Printemps [START_REF] Fournier | Absolute continuity for some one-dimensional processes[END_REF] for weaker assumptions on b), moreover this density admits a derivative with respect to the parameter θ.

F (z) = 1 |z| α+1 τ (z), (3)
We are interested in the statistical properties of the process (X θ t ), based on the discrete time observations (X θ i n ) i=0,...,n . Before stating our main results, we introduce some more notations. We denote by p θ 1 n (x, y) the transition density of the homogenous Markov chain (X θ i n ) i=0,...,n and by P θ n the law of the vector (X θ 1 n , . . . , X θ 1 ). In all the paper, for a function f depending on both variables (x, θ), we denote by f the derivative of f with respect to the variable x and by ḟ the derivative of f with respect to the parameter θ.

We first give an asymptotic expansion of log-likelihood ratio.

Theorem 1 Let u n = n 1 2 -1 α . We have : log dP θ+unh n dP θ n (X θ 1 n , . . . , X θ 1 ) = hJ n (θ) 1 2 N n (θ) - h 2 2 J n (θ) + o P (1), (4)
with :

J n (θ) = u 2 n n-1 i=0 E (ξ θ i,n ) 2 |G i/n N n (θ) = J n (θ) -1 2 u n n-1 i=0 ξ θ i,n ξ θ i,n = ṗθ 1 n p θ 1 n (X θ i n , X θ i+1 n
).

We can precise the asymptotic behavior of J n (θ) and N n (θ). Let ϕ α be the density of L α 1 , where (L α t ) is a centered α-stable Lévy process whose Lévy measure is dz |z| 1+α . We define the following quantity which will be the asymptotic information of the statistical model :

I θ = 1 0 ḃ(X θ s , θ) 2 ds × R ϕ α (u) 2 ϕ α (u) du. ( 5
)
Theorem 2 With the notations of Theorem 1, the following convergences hold :

J n (θ) n→∞ ---→ I θ , in probability, (6)
∀ε > 0, n-1 i=0 u 2 n E ξ θ i,n 2
1 {un|ξ θ i,n |≥ε} n→∞ ---→ 0. ( 7
)
Theorem 3 We have the convergence in law

J n (θ) 1 2 N n (θ) = u n n-1 i=0 ξ θ i,n n→∞ ---→ N (0, I θ ) , (8)
where the limit variable is conditionally Gaussian (recall the definition of I θ (5)), and the convergence is stable with respect to G 1 .

The stable convergence in law [START_REF] Ishikawa | Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps[END_REF] and the convergence in probability (6) yield to the convergence of the couple (J n (θ), N n (θ)):

(J n (θ), N n (θ)) n→∞ ---→ (I θ , N ), in law,
where N is a standard gaussian variable independent of I θ .

As a consequence of the asymptotic expansion given in Theorem 1 and the preceding limit theorems, we deduce the LAMN property. given by [START_REF] Genon | On the estimation of the diffusion coefficient for multidimensional diffusion processes[END_REF].

Let us stress that the rate of convergence depends on α. When α tends to 2, the rate u n degenerates.

This reflects the situation of a stochastic differential equation driven by a Brownian motion, where the drift coefficient cannot be estimated from the observation of the process on a finite time interval.

The proves of Theorem 1, Theorem 2 and Theorem 3 will be given in the next sections. They rely on the pointwise convergence of the transition density p θ 1 n (x 0 , y) and its derivative with respect to θ that will be study in Section 4. These asymptotic behaviors are precised below, after a time rescaling.

Let q n,θ,x 0 be the density of the rescaled variable n

1 α (X θ 1/n -x 0 ). One can verify that n 1 α (X θ 1/n -x 0 ) is equal in law to Y n,θ,x 0 1 solution of the equation Y n,θ,x 0 t = n 1 α -1 t 0 b(x 0 + n -1/α Y n,θ,x 0 s , θ)ds + L n,α t , (9)
with (L n,α t ) equal in law to (n 1/α L t/n ). The connection between the two densities is given by :

p θ 1/n (x 0 , y) = n 1 α q n,θ,x 0 (n 1 α (y -x 0 )). ( 10
)
The next result precises the asymptotic behavior of q n,θ,x 0 and qn,θ,x 0 as well as the limit of the Fisher information carried by the observation of Y n,θ,x 0 1 ,

I n,θ,x 0 = E   qn,θ,x 0 (Y n,θ,x 0 1 ) q n,θ,x 0 (Y n,θ,x 0 1 ) 2   . (11)
.

Proposition 1 For all (x 0 , u) ∈ R 2 , we have

i) q n,θ,x 0 (u) n→∞ ---→ ϕ α (u), ii) n 1-1/α qn,θ,x 0 (u) n→∞ ---→ ḃ(x 0 , θ)ϕ α (u), iii) n 2-2/α I n,θ,x 0 n→∞ ---→ ḃ(x 0 , θ) 2 E ϕ α (L α 1 ) ϕα(L α 1 ) 2 = ḃ(x 0 , θ) 2 R ϕ α (u) 2 ϕα(u) du.
The proof of this convergence result is based on the representation of the density q n,θ,x 0 and its derivative using the Malliavin calculus for jump processes. This is developed in the next sections.

Representation of the transition density via Malliavin calculus

The aim of this section is to represent q n,θ,x 0 and qn,θ,x 0 q n,θ,x 0 as an expectation, using the Malliavin calculus for jump processes developped by Bichteler, Gravereaux and Jacod [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF]. Due to the singularity of the Lévy measure of (L t ) at zero, we are not exactly in the same context, and we define in the next section an integration by part setting adapted to the study of equation ( 9).

Integration by part setting

In this section, we consider a filtered probability space (Ω, G, (G t ) t∈[0,1] , P ) endowed with a Poisson random measure µ on [0, 1] × E, where E is an open subset of R, with compensator ν given by dν = dt × g(z)dz on [0, 1] × E. We denote by μ the compensated measure and we are interested to study the regularity of the density of Y θ 1 , where the process (Y θ t ) is solution of :

Y θ t = t 0 a(Y θ s , θ)ds + t 0 E z μ(ds, dz). (12)
This is the framework of Bichteler, Gravereaux and Jacod [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF], excepted that g is not assumed to be equal to one and consequently the Malliavin operators have to be defined accordingly.

We make the following assumptions.

H: a) We assume that a is bounded with bounded derivatives up to order three with respect to both variables.

b) We assume that g ≥ 0 on E, C 1 on E and that

∀p ≥ 2, E |z| p g(z)dz < ∞.
We first precise the Malliavin operators L and Γ and their basic properties (see Bichteler, Gravereaux, Jacod, [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] Chapter IV, sections 8-9-10). For a test function f : [0, 1] × E → R ( f is measurable, C 2 with respect to the second variable, with bounded derivatives, and f ∈ ∩ p≥1 L p (ν)), we set

µ(f ) = 1 0 E f (t, z)µ(dt, dz).
We introduce an auxiliary function ρ : E → (0, ∞), derivable and such that ρ, ρ and ρ g g belong to ∩ p≥1 L p (g(z)dz). With these notations, we define the Malliavin operator L as

L(µ(f )) = 1 2 µ ρ f + ρ g g f + ρf , (13)
where f and f are the derivatives with respect to the second variable. For φ = F (µ(f 1 ), . . . , µ(f k )), with F of class C 2 , we set

LΦ = k i=1 ∂F ∂x i (µ(f 1 ), . . . , µ(f k )) L(µ(f i )) + 1 2 k i,j=1 ∂ 2 F ∂x i ∂x j (µ(f 1 ), . . . , µ(f k ))µ(ρf i f j ). ( 14
)
These definitions permit to construct a linear operator on a space D ⊂ ∩ p≥1 L p whose basic properties are the following :

i) L is self-adjoint : ∀Φ, Ψ ∈ D, we have EΦLΨ = ELΦΨ. ii) LΦ 2 ≥ 2ΦLΦ.
iii) ELΦ = 0.

We associate to L, the symmetric bilinear operator Γ :

Γ(Φ, Ψ) = L(ΦΨ) -ΦLΨ -ΨLΦ. ( 15
)
If f and h are two test functions, we have :

Γ(µ(f ), µ(h)) = µ ρf h , (16)
This operator satisfies the chain rule property :

Γ(F (Φ), Ψ) = F (Φ)Γ(Φ, Ψ). ( 17
)
Moreover we have the inequality

|Γ(Φ, Ψ)| ≤ Γ(Φ, Φ) 1/2 Γ(Ψ, Ψ) 1/2 . ( 18
)
These operators permit to establish the following integration by part formula (see [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] Proposition 8-10 p. 103).

Proposition 2 For Φ and Ψ in D, and f bounded with bounded derivatives up to order two, we have

Ef (Φ)ΨΓ(Φ, Φ) = Ef (Φ)(-2ΨLΦ -Γ(Φ, Ψ)).
Moreover, if Γ(Φ, Φ) is invertible and Γ -1 (Φ, Φ) ∈ ∩ p≥1 L p , we have

Ef (Φ)Ψ = Ef (Φ)H Φ (Ψ), (19)
with

H Φ (Ψ) = -2ΨΓ -1 (Φ, Φ)LΦ -Γ(Φ, ΨΓ -1 (Φ, Φ)). ( 20
)

Representation of the density of Y θ 1

The integration by part setting of the preceding section permits to derive the existence of the density of Y θ 1 given by ( 12), and gives a representation of this density as an expectation. Following Bichteler, Gravereaux, Jacod [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] (section 10, p130), we can prove that ∀t > 0, the variable Y θ t , solution of ( 12), belongs to the domain of the operator L, and we can compute LY θ t and Γ(Y θ t , Y θ t ).

Lemma 1 There are versions of the processes (LY θ t ) t∈[0,1] and (U θ t ) t = (Γ(Y θ t , Y θ t )) t that are solutions of the linear equations:

LY θ t = t 0 a (Y θ s , θ)LY θ s ds + 1 2 t 0 a (Y θ s , θ)U θ s ds + 1 2 t 0 E ρ (z) + ρ(z) g (z) g(z) µ(ds, dz), (21)
U θ t = 2 t 0 a (Y θ s , θ)U θ s ds + t 0 E ρ(z)µ(ds, dz). ( 22
)
The proof of this result is based essentially on the linearity and chain rule property of the operators L and Γ.

Theorem 4 Let us denote by q θ the density of Y θ 1 . We assume that the auxiliary function ρ satisfies:

lim inf u→∞ 1 ln u E 1 {ρ(z)≥1/u} g(z)dz = +∞. ( 23
)
Then we have :

q θ (u) = E(1 {Y θ 1 ≥u} H θ (1)), (24)
with

H θ (1) := H Y θ 1 (1) = Γ(Y θ 1 , Γ(Y θ 1 , Y θ 1 )) Γ(Y θ 1 , Y θ 1 ) 2 -2 LY θ 1 Γ(Y θ 1 , Y θ 1 ) (25)
Remark 1

The assumption (23) is a non degeneracy assumption which ensures the existence and

integrability of Γ(Y θ 1 , Y θ 1 ) -1 .
Proof We apply the integration by part formula (19) with f a regularization of the Dirac mass, Φ = Y θ 1 and Ψ = 1. So we just have to verify that, assuming (23

), U θ 1 = Γ(Y θ 1 , Y θ 1 ) is invertible and that 1 U θ 1 ∈ ∩ p≥1 L p .
From Lemma 1, solving equation ( 22), we obtain

U θ 1 = e 1 0 2a (Y θ s ,θ)ds 1 0 E e -s 0 2a (Y θ u ,θ)du ρ(z)µ(ds, dz).
Since a is bounded and ρ > 0, we deduce:

U θ 1 ≥ C 1 0 E ρ(z)µ(ds, dz),
where C is a non negative constant. We set I t (ρ) = t 0 E ρ(z)dµ(s, z), and we just have to prove that ∀p ≥ 1,

E 1 I 1 (ρ) p < ∞. We remark that ∀λ > 0, p ≥ 1, 1 λ p = C p ∞ 0 u p-1 e -λu du,
where C p is a non negative constant depending on p. So we deduce from Fubini Theorem :

E 1 I 1 (ρ) p = C p ∞ 0 u p-1 E(e -uI 1 (ρ) )du.
But from the classical exponential formula for Poisson measures, we have

Ee -uI 1 (ρ) = e -E (1-e -uρ(z) )g(z)dz .
We finally obtain:

E 1 I 1 (ρ) p = C p ∞ 0 u p-1 e -E (1-e -uρ(z) )g(z)dz du.
From assumption (23), we conclude easily that

E 1 I 1 (ρ) p < ∞.
To complete the result of Theorem 4, we give an expression for Γ(

Y θ t , Γ(Y θ t , Y θ t )).
Lemma 2 There is a version of (W θ t ) t = (Γ(Y θ t , U θ t )) t which is solution of the linear equation :

W θ t = 3 t 0 a (Y θ s , θ)W θ s ds + 2 t 0 a (Y θ s , θ)(U θ s ) 2 ds + t 0 E ρ(z)ρ (z)µ(ds, dz). ( 26
)
U θ s [ ȧ (Y θ s , θ) + a (Y θ s , θ) Ẏ θ s ]ds. ( 29
)
Proof Let f be a smooth function, by differentiating θ → Ef (Y θ 1 ), and using the integration by part formula (19), we obtain

f (u) qθ (u)du = Ef (Y θ 1 ) Ẏ θ 1 , = Ef (Y θ 1 )H θ ( Ẏ θ 1 ) = Ef (Y θ 1 )E(H θ ( Ẏ θ 1 )|Y θ 1 ) = f (u)E(H θ ( Ẏ θ 1 )|Y θ 1 = u)q θ (u)du.
This gives the expression of qθ q θ (u). The expression of the weight H θ ( Ẏ θ 1 ) follows from (20) and the basic properties of the operator Γ. The expression (29) follows from [START_REF] Kawai | On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling[END_REF] and the fact that Ẏ θ is

solution to Ẏ θ t = t 0 {a (Y θ s , θ) Ẏ θ s + ȧ(Y θ s , θ)}ds.

Application to the representation of the density of the rescaled process

We apply the preceding results to study the asymptotic behavior of q n,θ,x 0 , and qn,θ,x 0 q n,θ,x 0 , as n goes to infinity, where q n,θ,x 0 is the density of Y n,θ,x 0 1 defined by [START_REF] Jacod | The Euler scheme for Lévy driven stochastic differential equations: limit theorems[END_REF].

We can observe that process (L n,α t ), governing [START_REF] Jacod | The Euler scheme for Lévy driven stochastic differential equations: limit theorems[END_REF], and equal in law to (n 1/α L t/n ), is a centered Lévy process with Lévy measure

F n (z) = 1 |z| 1+α τ ( z n 1/α )
where τ is a non negative function equal to 1 on [-1, 1], vanishing on [-2, 2] c and satisfying 0 ≤ τ ≤ 1. This clearly suggests that when n growths, the process (L n,α t ) t becomes close to an α-stable process. For the sequel, it will be convenient to construct the family of Lévy processes (L n,α t ) t , for n ≥ 1, on a common probability space where the limiting α-stable process exists as well, and where the convergence holds true in a pathwise sense.

Let us consider µ e (dt, dz, du) a Poisson measure on [0, ∞) × R * × [0, 1] with compensating measure ν e (dt, dz, du) = dt dz |z| 1+α du. This measure corresponds to the jump measure of an α-stable process, where each jumps is marked with a uniform variable on [0, 1].

We define the Poisson measures µ (n) , for all n ≥ 1, and µ by setting:

∀A ⊂ [0, ∞) × R, µ (n) (A) = [0,∞) R [0,1] 1 A (t, z)1 {u≤τ ( z n 1/α )} µ e (dt, dz, du), ( 30
) ∀A ⊂ [0, ∞) × R, µ(A) = [0,∞) R [0,1] 1 A (t, z)µ e (dt, dz, du). ( 31
)
By simple computations, one can check that the compensator of the measure

µ (n) (dt, dz) is ν (n) (dt, dz) = dt × τ ( z n 1/α ) dz |z| 1+α = dt × F n (z)dz and the compensator of µ(dt, dz) is ν(dt, dz) = dt × dz |z| 1+α .
Remark that, since τ (z) = 1 for |z| ≤ 1, the measures µ (n) (ds, dz) and µ(ds, dz) coincides on the set

{(s, z) | |z| ≤ n 1/α }.
We now define the stochastic processes associated to these random measures,

L α t = t 0 [-1,1] z{µ(ds, dz) -ν(ds, dz)} + t 0 [-1,1] c zµ(ds, dz) (32) L n,α t = t 0 R z{µ (n) (ds, dz) -ν (n) (ds, dz)} = t 0 |z|≤2n 1/α z{µ (n) (ds, dz) -ν (n) (ds, dz)} (33)
By construction, the process L α is a centered α-stable process,and the process L n,α is equal in law to the process (n 1/α L t/n ) t , since they are based on random measures with same compensators. Remark that the jumps of L n,α t with size smaller than n 1/α exactly coincide with the jumps of L α with size smaller than n 1/α . On the other hand, the process L n,α has no jump with a size greater than 2n 1/α .

Using that the measures µ and µ (n) coincide on the subsets of {(t, z); |z| ≤ n 1/α }, and that, on

|z| ≤ n 1/α , the function τ ( z n 1/α ) 1 |z| 1+α = 1 |z| 1+α is symmetric, we can write: L n,α t = t 0 [-1,1] z{µ(ds, dz) -ν(ds, dz)} + t 0 1≤|z|≤n 1/α zµ(ds, dz) + t 0 n 1/α ≤|z|≤2n 1/α z{µ n (ds, dz) -ν n (ds, dz)}. ( 34
)
The following simple lemma gives a precise connection between L n,α and the stable process L α .

Lemma 3 There exists a sequence κ n with κ n n→∞ ---→ 0 such that for all t ≤ 1,

L n,α t = L α t -tκ n (35) on the event µ {(t, z) | 0 ≤ t ≤ 1, |z| ≥ n 1/α } = 0. Moreover P µ {(t, z) | 0 ≤ t ≤ 1, |z| ≥ n 1/α } = 0 = 1 + O(1/n). ( 36
)
Proof Let us set κ n = n 1/α ≤|z|≤2n 1/α zτ (z/n 1/α ) dz |z| 1+α which converges to zero since τ is bounded and α > 1. Now, by comparison of the representations (32) and (34), it is clear that the equation (35) holds true on the event that the supports of the random measures µ and µ (n) do not intersect

{(t, z) | 0 ≤ t ≤ 1, |z| ≥ n 1/α }. Since, by construction, the support of µ (n) is included in the support of µ, we see that (35) holds true on the event µ {(t, z) | 0 ≤ t ≤ 1, |z| ≥ n 1/α } = 0.
Finally, the probability of the latter event is exp -

1 0 |z|≥n 1/α
dz |z| 1+α dt which converges to 1 at rate 1/n as stated.

In the following, we will assume that the process Y n,θ,x 0 is solution of

Y n,θ,x 0 t = n 1 α -1 t 0 b(x 0 + n -1/α Y n,θ,x 0 s , θ)ds + L n,α t , (37)
where L n,α t is given by (33). We are in the framework of section 3.2, with g(z) = 1 |z| 1+α τ ( z n 1/α ) and the auxiliary function ρ can be chosen as ρ(z) = z 4 τ (2z).

Proposition 3 Let q n,θ,x 0 be the density of Y n,θ,x 0 1 , we have :

q n,θ,x 0 (u) = E(1 {Y n,θ,x 0 1 ≥u} H n θ (1)), ( 38
)
with

H n θ (1) := H Y n,θ,x 0 1 (1) = H n θ (1) + R n θ (1). ( 39
)
The main term H n θ (1) is given by

H n θ (1) = 1 0 R (E n s ) -3 ρ (z)ρ(z)µ(ds, dz) E n 1 1 0 R (E n s ) -2 ρ(z)µ(ds, dz) 2 - 1 0 R (E n s ) -1 ρ (z) -(1+α)ρ(z) z µ(ds, dz) E n 1 1 0 R (E n s ) -2 ρ(z)µ(ds, dz) (40)
where

E n t = exp n -1 t 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)ds , (41)
and the remainder term satisfies the upper bound,

|R n θ (1)| ≤ C 1 n 1+ 1 α , ( 42
)
where C is some deterministic constant.

Proof

We apply the Integration by Part Formula given in Theorem 4 to Y n,θ,x 0

1

. The non degeneracy assumption is verified by choosing ρ(z) = z 4 τ (2z). We obtain :

q n,θ,x 0 (u) = E(1 {Y n,θ,x 0 1 ≥u} H n θ (1)),
with

H n θ (1) = Γ(Y n,θ,x 0 1 , Γ(Y n,θ,x 0 1 , Y n,θ,x 0 1 )) Γ 2 (Y n,θ,x 0 1 , Y n,θ,x 0 1 ) -2 LY n,θ,x 0 1 Γ(Y n,θ,x 0 1 , Y n,θ,x 0 1 )
.

The random variables appearing in the weight H n θ (1) can be computed explicitely. Let us denote by

U n,θ t = Γ[Y n,θ,x 0 t , Y n,θ,x 0 t ], and W n,θ t = Γ[Y n,θ,x 0 t , U n,θ t ].
Then applying the results of Lemma 1 and Lemma 2 we have,

U n,θ t = 2 n t 0 U n,θ s b (x 0 + n -1/α Y n,θ,x 0 s , θ)ds + t 0 R ρ(z)µ (n) (ds, dz), L(Y n,θ,x 0 t ) = 1 n t 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)L(Y n,θ,x 0 s )ds + 1 2n 1+1/α t 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)U n,θ s ds + 1 2 t 0 R (ρ (z) + ρ(z) F n (z) F n (z) )µ (n) (ds, dz), W n,θ t = 3 n t 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)W n,θ s ds + 2 n 1+1/α t 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)(U n,θ s ) 2 ds + t 0 R ρ(z)ρ (z)µ (n) (ds, dz).
These linear equations can be resolved explicitly using E n t given by (41). By simple computations, we find

U n,θ 1 = (E n 1 ) 2 1 0 R (E n s ) -2 ρ(z)µ(ds, dz), ( 43
)
where we used that the measures µ (n) (ds, dz) and µ(ds, dz) coincide on (s, z)

∈ [0, 1] × [-1, 1] and that the support of ρ is included in [-1, 1]
. By analogous computations we get,

L(Y n,θ,x 0 1 ) = E n 1 2 1 0 R (E n s ) -1 ρ (z) - (1 + α)ρ(z) z µ(ds, dz) + E n 1 2n 1+1/α 1 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)U n,θ s (E n s ) -1 ds (44)
where we have used that F n (z) Fn(z) = -1+α z on the support of ρ. Solving the equation for W n,θ 1 yields to

W n,θ 1 = (E n 1 ) 3 1 0 R (E n s ) -3 ρ (z)ρ(z)µ(ds, dz) + (E n 1 ) 3 2n 1+1/α 1 0 b (x 0 + n -1/α Y n,θ,x 0 s , θ)(U n,θ s ) 2 (E n s ) -3 ds. ( 45
)
Based on these expressions and recalling that

H n θ (1) = W n,θ 1 (U n,θ 1 ) 2 -2 L(Y n,θ,x 0 1 ) U n,θ 1 , (46)
we deduce, after some calculus, the decomposition (39), where the leading term is

H n θ (1) = 1 0 R (E n s ) -3 ρ (z)ρ(z)µ(ds, dz) E n 1 1 0 R (E n s ) -2 ρ(z)µ(ds, dz) 2 - 1 0 R (E n s ) -1 ρ (z) -(1+α)ρ(z) z µ(ds, dz) E n 1 1 0 R (E n s ) -2 ρ(z)µ(ds, dz)
and, using that b is bounded with bounded derivatives and U n,θ s U n,θ 1 is bounded for 0 ≤ s ≤ 1, the remainder term satisfies the upper bound

|R n θ (1)| ≤ C n 1+1/α
where C is some deterministic constant.

On a similar way, we give an expansion of qn,θ,x 0 q n,θ,x 0 (u).

Proposition 4

We have :

qn,θ,x 0 q n,θ,x 0 (u) = E(H n θ ( Ẏ n,θ,x 0 1 )|Y n,θ,x 0 1 = u), (47)
with

H n θ ( Ẏ n,θ,x 0 1 ) := H Y n,θ,x 0 1 ( Ẏ n,θ,x 0 1 ) = Ẏ n,θ,x 0 1 H n θ (1) + R n θ ( Ẏ n,θ,x 0 1 ), ( 48
)
where R n θ ( Ẏ n,θ,x 0

1

) ≤ Cn -1 and H n θ (1) is given in (40).

Proof Using successively the Theorem 5 and the equation (46), we have

H n θ ( Ẏ n,θ,x 0 1 ) = Ẏ n,θ,x 0 1 W n,θ 1 (U n,θ 1 ) 2 -2 LY n,θ,x 0 1 U n,θ 1 - Γ(Y n,θ,x 0 1 , Ẏ n,θ,x 0 1 ) U n,θ 1 = Ẏ n,θ,x 0 1 H n θ (1)- Γ(Y n,θ,x 0 1 , Ẏ n,θ,x 0 1 ) U n,θ 1 , U n,θ
1 is given by (43). For the computation of V n,θ

1 = Γ(Y n,θ,x 0 1 , Ẏ n,θ,x 0 1
), we use (29), this gives

V n,θ 1 = (E n 1 ) 2 1 0 (E n s ) -2 U n,θ s 1 n ḃ (x 0 + n -1/α Y n,θ,x 0 s , θ) + Ẏ n,θ,x 0 s 1 n 1+1/α b (x 0 + n -1/α Y n,θ,x 0 s , θ) ds. (49)
The expression of Ẏ n,θ,x 0 1 is explicitly given by,

Ẏ n,θ,x 0 1 = n 1 α -1 E n 1 1 0 (E n s ) -1 ḃ(x 0 + n -1/α Y n,θ,x 0 s , θ)ds. ( 50
)
Using these expressions, we deduce the bounds

Ẏ n,θ,x 0 1 ≤ Cn 1 α -1 (51) V n,θ 1 ≤ Cn -1 U n,θ 1
Combining this with the Proposition 3, the result follows.

Asymptotic behaviour of the transition density

In this section we study the asymptotic behaviour of q n,θ,x 0 , the density of Y n,θ,x 0 1 , solution of (37).

We will establish some stronger versions of Proposition 1.

Pointwise convergence

The following two propositions will imply the results of Proposition 1 i) and ii).

Proposition 5 Let (θ n ) n≥1 be a sequence of parameters such that θ n n→∞ ---→ θ. For all (x 0 , u) ∈ R 2 ,
we have q n,θn,x 0 (u)

n→∞ ---→ ϕ α (u). Moreover sup u∈R sup n q n,θn,x 0 (u) < ∞. ( 52
) Proposition 6 Let (θ n ) n≥1 be a sequence of parameters such that θ n n→∞ ---→ θ. For all (x 0 , u) ∈ R 2 , we have √ nu n qn,θn,x 0 (u) n→∞ ---→ ḃ(x 0 , θ)ϕ α (u). Moreover sup u sup n n 1-1/α qn,θn,x 0 (u) < ∞. (53)

Proof of the Proposition 5

From the Proposition 3, the expression for the density of Y n,θ,x 0 1 at some point u and with θ n ∈ Θ is given by equation ( 38)

q n,θn,x 0 (u) = E[1 [u,∞) (Y n,θn,x 0 1 )H n θn (1)],
where H n θn (1) = H n θn (1) + R n θn [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF], with H n θn [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF] given by (40) and R n θn (1) bounded by (42). Let us note

H L α (1) = 1 0 R ρ (z)ρ(z)µ(ds, dz) 1 0 R ρ(z)µ(ds, dz) 2 - 1 0 R ρ (z) -(1+α)ρ(z) z µ(ds, dz) 1 0 R ρ(z)µ(ds, dz) , ( 54
)
then from (41), and the boundedness of b , it is clear that H n θn (1) converges almost surely to H L α (1). Using again the boundedness of b and the fact that ρ is a non negative function, we deduce the upper bound

H n θn (1) ≤ C 1 0 R |ρ (z)| ρ(z)µ(ds, dz) 1 0 R ρ(z)µ(ds, dz) 2 + 1 0 R |ρ (z)| + (1+α)ρ(z) |z| µ(ds, dz) 1 0 R ρ(z)µ(ds, dz) (55)
for some constant C > 0. Using that ρ, |ρ | and z → ρ(z) z belongs to p≥1 L p (|z| -1-α dz) we get

E 1 0 R |ρ (z)| ρ(z)µ(ds, dz) p < ∞, E 1 0 R |ρ (z)| + (1+α)ρ(z) |z| µ(ds, dz) p < ∞ for all p ≥ 1.
Since ρ satisfies the non degeneracy assumption (23), [ 1 0 R ρ(z)µ(ds, dz)] -1 belongs to p≥1 L p , as a consequence we deduce from (55) that sup n H n θn [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF] p is integrable for all p ≥ 1. Applying the dominated convergence Theorem, we deduce that

H n θn (1) n→∞ ---→ L p H L α (1), , ∀p ≥ 1. ( 56
)
The Lemma 3 implies that L n,α 1 converges to L α 1 in probability. From the boundedness of b and equation (37) we deduce that Y n,θn,x 0 1 converges in probability to L α 1 . Then, an easy computation, using that P (L α 1 = u) = 0, shows the convergence in probability

1 [u,∞) (Y n,θn,x 0 1 ) n→∞ ---→ P 1 [u,∞) (L α 1 ). ( 57
)
Moreover from the boundedness property of the variables, the latter convergence holds in L p sense, ∀p ≥ 1.

Using (38), (39), ( 42), ( 56), (57) we get,

q n,θn,x 0 (u) n→∞ ---→ E 1 [u,∞) (L α 1 )H L α (1) . (58)
To finish the proof of the convergence, it remains to show that the right hand side of (58) is a representation for ϕ α (u), the density of an α-stable process. This is done in Lemma 4 below.

Remark that, we easily get from (38), (39), (42), and (56) that sup u∈R sup n q n,θn,x 0 (u) < ∞.

Lemma 4

We have

ϕ α (u) = E 1 [u,∞) (L α 1 )H L α (1) . ( 59
)
Proof The relation (59) could be formally obtained by Malliavin computations for the stable measure.

However Malliavin computation in the setting of a stable process does not immediately enters the framework developed by [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF], since the amplitude of the (big) jumps are not L p random variables for all p. Hence, we prefer to give another proof.

Let us denote ϕ n (u) the density of the variable L n,α 1 . We apply the results ( 52) and (58), in the situation where the drift function b ≡ 0, for which Y n,θ,x 0 1 = L n,α 1 . This yields to,

ϕ n (u) n→∞ ---→ E 1 [u,∞) (L α 1 )H L α (1) := ψ(u), (60)
sup u sup n ϕ n (u) < ∞. (61)
Assume by contradiction that, for some u, we have ψ(u) = ϕ α (u). From the fact that P (L α 1 = u) = 0, it can be seen that ψ is continuous at the point u. Hence, one can find a continuous, compactly supported, function f such that f (x)ψ(x)dx = f (x)ϕ α (x)dx.

On the one hand we have,

E[f (L n,α 1 )] = f (x)ϕ n (x)du n→∞ ---→ f (x)ψ(x)dx, (62)
where we have used the dominated convergence Theorem with (60)-(61). On the other hand, we write

E[f (L n,α 1 )] = E[f (L α 1 -κ n )1 {L n,α 1 =L α 1 -κn} ] + E[f (L n,α 1 )1 {L n,α 1 =L α 1 -κn} ],
where we have used the notations of Lemma 3. Moreover, by Lemma 3, we have P (L n,α

1 = L α 1 - κ n ) n→∞ ---→ 1. We deduce that, E[f (L n,α 1 )] n→∞ ---→ E[f (L α 1 )] = f (x)ϕ α (x)dx (63)
This last convergence result clearly contradicts (62).

Proof of Proposition 6

First we write a representation as an expectation for qn,θ,x 0 . Let f be a smooth, non negative and compactly supported function. Differentiating the relation E[f (Y n,θ,x 0

1

)] = f (u)q n,θ,x 0 du, we get,

E[f (Y n,θ,x 0 1 ) Ẏ n,θ,x 0 1 ] = f (u) qn,θ,x 0 (u)du.
Using the Integration by Part Formula (19) , we obtain

f (u) qn,θ,x 0 (u)du = E[f (Y n,θ,x 0 1 )H n θ ( Ẏ n,θ,x 0 1 )], (64)
where

H n θ ( Ẏ n,θ,x 0 1
) is given by ( 48) :

H n θ ( Ẏ n,θ,x 0 1 ) = Ẏ n,θ,x 0 1 H n θ (1) + R n θ ( Ẏ n,θ,x 0 1 ), with R n θ ( Ẏ n,θ,x 0 1
) ≤ Cn -1 and H n θ (1) given in (40). Using (64), we get

f (u) qn,θ,x 0 (u)du -E[f (Y n,θ,x 0 1 ) Ẏ n,θ,x 0 1 H n θ (1)] ≤ E f (Y n,θ,x 0 1 ) R θ ( Ẏ n,θ,x 0 1 ) ≤ Cn -1 E[f (Y n,θ,x 0 1 )]
Applying the Integration by Part Formula, we deduce

f (u) qn,θ,x 0 (u)du -E F (Y n,θ,x 0 1 )H Y n,θ,x 0 1 Ẏ n,θ,x 0 1 H n θ (1) ≤ Cn -1 E[f (Y n,θ,x 0 1 )]
where F denotes a primitive function of f and

H Y n,θ,x 0 1 Ẏ n,θ,x 0 1 H n θ (1)
is defined by (20). If f converges to a Dirac mass at some point u, we deduce, qn,θ,x 0 (u)

-E 1 [u,∞) (Y n,θ,x 0 1 )H Y n,θ,x 0 1 Ẏ n,θ,x 0 1 H n θ (1) ≤ Cn -1 q n,θ,x 0 (u). (65)
Thus we need to study lim

n n 1-1/α E 1 [u,∞) (Y n,θn,x 0 1 )H Y n,θn,x 0 1 Ẏ n,θn,x 0 1
H n θn [START_REF] Aït | Volatility estimators for discretely sampled Lévy processes[END_REF] . Actually, the main step is to show that

n 1-1/α H Y n,θn,x 0 1 Ẏ n,θn,x 0 1 H n θn (1) n→∞ ---→ L p ḃ(x 0 , θ)H (2) , ∀p ≥ 1, ( 66
)
where H (2) is some random variable whose expression does not depend on θ and b. This is done in Lemma 10 (see the section 7). Then, as in the proof of (58), we can deduce from (65)-(66), that

n 1-1/α qn,θn,x 0 (u) n→∞ ---→ ḃ(x 0 , θ)E 1 [u,∞) (L α 1 )H (2) . (67)
Remark that from (52) and ( 65)-(66), we get sup

u sup n n 1-1/α qn,θn,x 0 (u) < ∞.
The proof of the Proposition will be finished if we identify E 1 [u,∞) (L α 1 )H (2) as being equal to ϕ α (u). This is done in Lemma 5 below.

Lemma 5

We have for all u ∈ R,

ϕ α (u) = E[1 [u,∞) (L α 1 )H (2) ]. ( 68
)
Proof Let us consider the situation where b(x, θ) = θ. In that case, we have

Y n,θ,x 0 1 = n 1 α -1 θ + L α,n 1
and thus the density of Y n,θ,x 0 1 is related to the density of L α,n 1 by the relation

q n,θ,x 0 (u) = ϕ n (u -n 1/α-1 θ).
We can apply the results ( 53) and (67) in this specific setting. This yields to

∀u, ϕ n (u -n 1/α-1 θ) n→∞ ---→ E[1 [u,∞) (L α 1 )H (2) ], (69) sup
u,n ϕ n (u -n 1/α-1 θ) < ∞. ( 70
) Let us denote χ(u) = E[1 [u,∞) (L α
1 )H (2) ] and assume by contradiction that χ = ϕ α . Using the continuity of u → χ(u), there exists a smooth, compactly supported function, f such that χ(u)f (u)du = ϕ α (u)f (u)du. Now, on the one hand we have

ϕ n (u -n 1/α-1 θ)f (u)du n→∞ ---→ χ(u)f (u)du, ( 71
)
where we have used the dominated convergence theorem, together with (69)-(70).

On the other hand, we can write

ϕ n (u -n 1/α-1 θ)f (u)du = -ϕ n (u -n 1/α-1 θ)f (u)du = -ϕ n (u)f (u + n 1/α-1 θ)du = -E[f (L α,n 1 + n 1-1/α )] n→∞ ---→ -E[f (L α 1 )] = -ϕ α (u)f (u)du (72) = ϕ α (u)f (u)du ( 73
)
where the convergence (72) is obtained in the same way as (63). Clearly (73) contradicts (71), and the lemma is proved.

Fisher information

We study now the asymptotic properties of the Fisher information defined by [START_REF] Jeganathan | Some asymptotic properties of risk functions when the limit of the experiment is mixed normal[END_REF] corresponding to the observation of the random variable Y n,θ,x 0

1

. We recall that it is given by

I n,θ,x 0 = E   qn,θ,x 0 (Y n,θ,x 0 1 ) q n,θ,x 0 (Y n,θ,x 0 1 ) 2  
We will show a stronger version of the Proposition 1 iii).

Proposition 7 Let (θ n ) be a sequence such that θ n n→∞ ---→ θ, we have i)

n 2-2/α I n,θn,x 0 n→∞ ---→ ḃ(x 0 , θ) 2 E ϕ α (L α 1 ) ϕ α (L α 1 ) 2 = ḃ(x 0 , θ) 2 R ϕ α (u) 2 ϕ α (u) du,
and this convergence is uniform with respect to

x 0 , ii) sup n,x 0 ,θ n 2-2/α I n,θ,x 0 = sup n,x 0 ,θ n 2-2/α E qn,θ,x 0 (Y n,θ,x 0 1 ) 2 q n,θ,x 0 (Y n,θ,x 0 1
) 2 < ∞.

The proof of this proposition is based on the following lemma, which is related to a continuity property with respect to the conditioning variable.

Lemma 6 Let (θ n ) n≥1 be a sequence such that θ n n→∞ ---→ θ. Then, the following convergence holds uniformly with respect to x 0 ,

n 2-2/α E E[H n θ ( Ẏ n,θn,x 0 1 ) | Y n,θn,x 0 1 ] 2 n→∞ ---→ ḃ(x 0 , θ) 2 E E[H L α (1) | L α 1 ] 2 ,
where H L α (1) is given by (54) and L α 1 by (32).

Proof Let us recall the crucial decomposition given in (48),

H n θ ( Ẏ n,θ,x 0 1 ) = Ẏ n,θ,x 0 1 H n θ (1)+R n θ ( Ẏ n,θ,x 0 1 ) where R n θ ( Ẏ n,θ,x 0 1 ) ≤ Cn -1 . From the fact that sup x 0 sup s∈[0,1] |E n s -1| + (E n s ) -1 -1 n→∞ ---→ 0 and
the expression explicit expression of Ẏ n,θ,x 0 1

given in (50), we easily get sup

x 0 n 1-1/α Ẏ n,θn,x 0 1 -ḃ(x, θ) n→∞ ---→ a.s.

0.

From the expression (40) and (54) it can be seen that sup

x 0 H n θn (1) -H L α (1) n→∞ ---→ a.s.
0. We deduce that almost surely, one has the convergence sup

x 0 n 1-1/α H n θn ( Ẏ n,θn,x 0 1 ) -ḃ(x 0 , θ)H L α (1) n→∞ ---→ 0. ( 74
)
Moreover using the upper bound (55) with (48) again, we can apply the dominated convergence Theorem and see that the convergence (74) holds in L p -norm for all p ≥ 1. Now, we can write

E E[n 1-1/α H n θn ( Ẏ n,θn,x 0 1 ) | Y n,θ,n,x 0 1 ] -E[ ḃ(x 0 , θ)H L α (1) | Y n,θ,n,x 0 1 ] 2 ≤ E E[ n 1-1/α H n θn ( Ẏ n,θn,x 0 1 ) -ḃ(x 0 , θ)H L α (1) 2 | Y n,θ,n,x 0 1 ] = E n 1-1/α H n θn ( Ẏ n,θn,x 0 1 ) -ḃ(x 0 , θ)H L α (1)
2 converges to zero uniformly with respect to x 0 . In turns, it gives the uniform convergence

n 2-2/α E E[H n θn ( Ẏ n,θn,x 0 1 ) | Y n,θn,x 0 1 ] 2 -ḃ(x 0 , θ) 2 E E[H L α (1) | Y n,θn,x 0 1 ] n→∞ ---→ 0.
Hence, the proposition will be proved as soon as we show the uniform convergence with respect to x 0 ,

E E[H L α (1) | Y n,θn,x 0 1 ] 2 -E E[H L α (1) | L α 1 ] 2 n→∞ ---→ 0. ( 75
)
This is a delicate part of the proof, since it amounts to compare the conditional expectation of a variable with respect to the two different variables Y n,θn,x 0 1 and L α 1 . First, we reduce the situation to the case where the random variable in the expectation is bounded. Let K > 0 and denote by x → χ K (x) a smooth truncation function with χ K (x) = 0 for |x| ≥ K, χ K (x) = 1 for |x| ≤ K/2 and 0 ≤ χ K ≤ 1. Using that E[H L α (1) 2 ] < ∞, one can see that (75) is implied by the following convergence for all K > 0, sup

x 0 E E[H L α (1)χ K (H L α (1)) | Y n,θn,x 0 1 ] 2 -E E[H L α (1)χ K (H L α (1)) | L α 1 ] 2 n→∞ ---→ 0. ( 76
)
Let us denote by η n and η the measurable functions such that,

E H L α (1)χ K (H L α (1)) | Y n,θn,x 0 1 = η n (Y n,θn,x 0 1
),

E [H L α (1)χ K (H L α (1)) | L α 1 ] = η(L α 1 ).
With these notations, the condition (76) writes sup

x 0 E[η n (Y n,θn,x 0 1 ) 2 ] -E[η(L α 1 ) 2 ] n→∞ ---→ 0 (77)
Using Proposition 10 in Section 7.2, we know that sup

x 0 E η(Y n,θn,x 0 1 ) -η n (Y n,θn,x 0 1 ) n→∞ ---→ 0.
Since |η n | and |η| are bounded by the constant K, we deduce η(Y n,θn,x 0 1 ) -η n (Y n,θn,x 0

---→ 0 uniformly with respect to x 0 . This yields to sup

x 0 E[η n (Y n,θn,x 0 1 ) 2 ] -E[η(Y n,θn,x 0 1 ) 2 ] n→∞ ---→ 0
Now, applying (104) in Corollary 2 with the bounded function η 2 yields to (77), and the lemma is proved.

We can now prove the main result of the Section.

Proof of Proposition 7. i) First we remark that we have the representation

ϕ α (u) ϕ α (u) = E H L α 1 (1) | L α 1 = u . (78)
Indeed, by considering the specific model b(x, θ) = θ, we obtain, Y n,θ,x 0

1 = L n,α 1 + n 1-1/α θ, Ẏ n,θ,x 0 1 = n 1-1/α , H n θ (Y n,θ,x 0 ) = n 1-1/α H L α (1), q n,θ,x 0 (u) = ϕ n (u -n 1-1/α θ).
Using (64), we get for any smooth function f ,

f (u)ϕ n (u -n 1-1/α θ)n 1-1/α du = E[f (L n,α 1 + n 1-1/α θ)n 1-1/α H L α (1)]
From the convergence results (73) and the smoothness of f , we get f (u)ϕ α (u)du = E[f (L α 1 )H L α (1)], and we deduce (78).

Next, we have from Proposition 4

n 2-2/α I n,θn,x 0 = n 2-2/α E E H n θn (Y n,θn,x 0 1 ) Y n,θn,x 0 2 , n→∞ ---→ ḃ(x 0 , θ) 2 E E[H L α (1) | L α 1 ] 2 , from Lemma 6, = ḃ(x 0 , θ) 2 E ϕ α (L α 1 ) 2 ϕ α (L α 1 ) 2 , from (78)
, which proves the first part of the proposition.

ii) Using successively the Proposition 4 and Jensen inequality, we get

I n,θ,x 0 = E[E[H n θ ( Ẏ n,θ,x 0 1 ) | Y n,θ,x 0 ] 2 ] ≤ E[H n θ ( Ẏ n,θ,x 0 1 ) 2 ]
But it is clear from (48), ( 51) and (55

) that n 1-1/α H n θ ( Ẏ n,θ,x 0 1
) is bounded in L p norm independently of n, θ, x 0 , for any p ≥ 1.

5 Proof of the asymptotic expansion of the likelihood (Theorems

1-2)
This section is devoted to the proof of the asymptotic expansion for the log-likelihood function, established in the Theorem 1. The proof is based essentially on the L 2 -regularity property of the transition density p θ 1/n (x, y) and on the result of Theorem 2. Indeed, from Jeganathan [START_REF] Jeganathan | On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal[END_REF], the following four conditions A1-A4 are sufficient to get the expansion (4) of Theorem 1.

We recall the notation ξ θ i,n = ṗθ

1 n p θ 1 n (X θ i n , X θ i+1 n ). A1. L 2 -regularity n j=1 E    R    p θ+unh 1 n (X θ j-1 n , y) 1/2 -p θ 1 n (X θ j-1 n , y) 1/2 - 1 2 hu n ṗθ 1 n (X θ j-1 y , y) p θ 1 n (X θ j-1 y , y) 1/2    2 dy    n→∞ ---→ 0. A2. J n (θ) = u 2 n n-1 i=0 E (ξ θ i,n ) 2 |G i/n n→∞ ---→ I θ (> 0 a.e.), in probability, A3. ∀ε > 0, n-1 i=0 u 2 n E ξ θ i,n 2
1 {un|ξ θ i,n |≥ε} n→∞ ---→ 0. A4. sup n u 2 n n i=0 E(ξ θ i,n ) 2 ≤ C, for a strictly positive constant C
The condition A1 is proved in Section 5.1 below. The conditions A2 and A3 coincide with the Theorem 2, which is proved in Section 5.2 below. The condition A4 is immediate from the Proposition

7 ii), since E(ξ θ i,n ) 2 = EI n,θ,X θ i/n and nu 2 n = n 2-2/α
. Note that these conditions does not imply the stable convergence in law [START_REF] Ishikawa | Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps[END_REF] since in our framework the filtration (G i n ) i does not satisfy the nested condition. The proof of the stable convergence in law will be given in Section 6.

Proof of the L 2 regularity condition

Proposition 8 Set u n = n 1/2-1/α , we have n j=1 E    R    p θ+unh 1 n (X θ j-1 n , y) 1/2 -p θ 1 n (X θ j-1 n , y) 1/2 - 1 2 hu n ṗθ 1 n (X θ j-1 y , y) p θ 1 n (X θ j-1 y , y) 1/2    2 dy    n→∞ ---→ 0. ( 79
)
Proof Recall that q n,θ,x 0 is the density of the rescaled process (X θ 1/n -x 0 )n 1/α . One has the simple relation p θ 1 n (x, y) = n 1/α q n,θ,x [n 1/α (y -x)], and proving (79) amounts to show the convergence to zero of the following quantity,

n j=1 E n 1/α R q n,θ+unh,X θ j-1 n (n 1/α (y -X θ j-1 n )) 1/2 -q n,θ,X θ j-1 n (n 1/α (y -X θ j-1 n )) 1/2 - 1 2 hu n qn,θ,X θ j-1 n (n 1/α (y -X θ j-1 n )) q n,θ,X θ j-1 n (n 1/α (y -X θ j-1 n )) 1/2 2 dy .
By a simple change of variable, it is equivalent to show

n -1 n j=1 E R n 1/2 [q n,θ+unh,X θ j-1 n (u) 1/2 -q n,θ,X θ j-1 n (u) 1/2 ] - 1 2 hn 1/2 u n qn,θ,X θ j-1 n (u) q n,θ,X θ j-1 n (u) 1/2 2 du n→∞ ---→ 0. (80)
Let us denote f n (x, u) = n 1/2 [q n,θ+unh,x (u) 1/2 -q n,θ,x (u) 1/2 ],

g n (x, u) = 1 2 n 1/2 u n h qn,θ,x (u) q n,θ,x (u) 1/2 .
Let us admit temporarily that the three following properties holds true :

1) There exists a function f such that,

∀x, u, f n (x, u) n→∞ ---→ f (x, u), g n (x, u) n→∞ ---→ f (x, u).
2) We have for all x,

lim sup n R f n (x, u) 2 du ≤ R f (x, u) 2 du, lim sup n R g n (x, u) 2 du ≤ R f (x, u) 2 du. 3) We have sup x,n R f n (x, u) 2 du < ∞, (81) sup
x,n R g n (x, u) 2 du < ∞. (82)
Admitting these three points, we can prove (80). Let ε > 0, we first show a uniform polynomial decay for y → p θ t (x 0 , y), when t ≥ ε. Using Theorem 4, we have

p θ t (x 0 , y) = E[1 {X θ t ≥y} H X θ t (1)],
where H X θ t

(1) is given by (20). But, it can be seen that sup ε≤t≤1 E[Γ(X θ t , X θ t ) -p ] is bounded for any p ≥ 1. Then, one can easily deduce that sup ε≤t≤1 E H X θ t (1) p < ∞. From (2) and the fact the Lévy measure of L has a compact support we deduce sup t∈[0,1] E[ X θ t p ] < ∞. Using the Markov inequality, we deduce that sup ε≤t≤1 p θ t (x 0 , y) ≤ C 1+y 2 for some C > 0. Then, we split the right hand side of (80) on the following way

n -1 n j=1 R E[{f n (X θ j-1 n , u) -g n (X θ j-1 n , u)} 2 ]du = n -1 nε j=1 E R {f n (X θ j-1 n , u) -g n (X θ j-1 n , u)} 2 du + n -1 n j= nε +1 E R {f n (X θ j-1 n , u) -g n (X θ j-1 n , u)} 2 du ≤ n -1 nε j=1 sup x,n R 2[f n (x, u) 2 + g n (x, u) 2 ]du + n -1 n j= nε +1 E R {f n (X θ j-1 n , u) -g n (X θ j-1 n , u)} 2 du ≤ εC + n -1 n j= nε +1 E R {f n (X θ j-1 n , u) -g n (X θ j-1 n , u)} 2 du , by (81)-(82), ≤ εC + n -1 C n j= nε +1 R R
{f n (y, u) -g n (y, u)} 2 du dy 1 + y 2 , using p θ t (x 0 , y) ≤

C 1 + y 2 , = εC + C n -nε n R R
{f n (y, u) -g n (y, u)} 2 du dy 1 + y 2 .

From Lemma 7, the conditions 1) and 2) imply that R {f n (y, u) -g n (y, u)} 2 du n→∞ ---→ 0. The condition 3) is sufficient to apply the dominated convergence Theorem and find that R R {f n (y, u)g n (y, u)} 2 du dy 1+y 2 converges to zero as n → ∞. Hence, we have proved the proposition, up to the fact that we need to check the validity of the conditions 1), 2) and 3).

We start with the proof of the property 1). From Propositions 5-6, we see that

g n (x, u) n→∞ ---→ f (x, u) := 1 2 h ḃ(x, θ) ϕ α (u) ϕα(u) 1/2 .
Using the mean value theorem, we can write

f n (x, u) = 1 2 n 1/2 u n h qn,θn,x (u)
q n,θn,x (u) 1/2 , for some θ n ∈ [θ, θ + u n h]. Using again the Propositions 5-6, we get f n (x, u) n→∞ ---→ f (x, u).

We now prove the property 2). Recalling that u n = n 1/2-1/α and (11), we have

R g n (x, u) 2 du = h 2
4 n 2-2/α I n,θ,x . From the Proposition 7, we get

R g n (x, u) 2 du n→∞ ---→ R f (x, u) 2 du, for all x.
Using f n (x, u) = 1 2 n 1/2 θ+unh θ qn,s,x (u) q n,s,x (u) 1/2 ds. We write

R f n (x, u) 2 du = n 4 θ+unh θ qn,s,x (•) q n,s,x (•) 1/2 ds 2 2 ≤ n 4 θ+unh θ qn,s,x (•) q n,s,x (•) 1/2 2 ds 2 = h 2 4 nu 2 n 1 0 R
qn,θ+sunh,x (u) 2 q n,θ+sunh,x (u) du

1/2 ds 2 = h 2 4 1 0 (n 2-2/α I n,θ+sunh,x ) 1/2 ds 2 (83) n→∞ ---→ h 2 4 ḃ(x, θ) 2 R ϕ α (u) 2 ϕ α (u) du = R f 2 (x, u)du,
where, in the last line, we have used the Proposition 7 for the convergence of n 2-2/α I n,θ+sunh,x and the application of the dominated convergence Theorem.

We end the proof of the Proposition by showing the property 3). From (83) and Proposition 7 we get (81). The bound (82) is deduced by Proposition 7 as well.

Lemma 7 Assume that (f n ) n , (g n ) n are two sequences of real functions such that:

1) There exists f ∈ L 2 (R) such that f n (u) n→∞ ---→ f (u) and g n (u) n→∞ ---→ f (u)
for almost every u.

2) We have lim sup

n R f n (u) 2 du ≤ R f (u) 2 du and lim sup n R g n (u) 2 du ≤ R f (u) 2 du. Then, R (f n (u) -g n (u)) 2 du n→∞ ---→ 0. Proof We write (f n (u)-g n (u)) 2 ≤ 2f n (u) 2 +2g n (u) 2 and thus 2f n (u) 2 +2g n (u) 2 -(f n (u)-g n (u)) 2 ≥ 0.
Applying Fatou's lemma to this non negative function, we get

R 4f (u) 2 du ≤ lim inf n R [2f n (u) 2 + 2g n (u) 2 -(f n (u) -g n (u)) 2 ]du ≤ lim sup n R [2f n (u) 2 + 2g n (u) 2 ]du -lim sup n R (f n (u) -g n (u)) 2 du.
This yields to the inequality lim sup

n R (f n (u)-g n (u)) 2 du ≤ lim sup n R 2f n (u) 2 du+lim sup n R 2g n (u) 2 du- R 4f (
u) 2 du ≤ 0, and thus the lemma follows.

Proof of Theorem 2

Proof First, we use that

ṗθ 1/n (x,y) p θ 1/n (x,y) = qn,θ,x (n 1/α (y-x))
q n,θ,x (n 1/α (y-x)) , and as a result of the Markov property for the process X θ and (11), we have

E[(ξ θ i,n ) 2 | G i/n ] = I n,θ,X θ i/n .
From the Proposition 7, we know that the quantity sup 0≤i≤n-1

nu 2 n I n,θ,X θ i/n -ḃ(X θ i/n , θ) 2 R ϕ (u) 2 ϕ(u) du = sup 0≤i≤n-1 n 2-2/α I n,θ,X θ i/n -ḃ(X θ i/n , θ) 2 R ϕ (u) 2 ϕ(u) du
converges to zero as n → ∞. Then the convergence ( 6) is a consequence of the convergence of a Riemann sum.

To prove [START_REF] Hájek | A characterization of limiting distributions of regular estimates[END_REF] we use, again, the relation

ṗθ 1/n (x,y) p θ 1/n (x,y) = qn,θ,x (n 1/α (y-x))
q n,θ,x (n 1/α (y-x)) and the Markov property to

get, E[|ξ i,n | k | X θ i/n = x] = E qn,θ,x (Y n,θ,x 1 ) q n,θ,x (Y n,θ,x 1 ) k , for any k ≥ 1. It then follows from Proposition 4 that, E[|ξ i,n | k | X θ i/n = x] = E E H n θ ( Ẏ n,θ,x 1 ) | Y n,θ,x 1 k ≤ E H n θ ( Ẏ n,θ,x 1 ) k ,
where we used the Jensen inequality in the last step. As seen in the proof of Proposition 7, the random

variables n 1-1/α H n θ ( Ẏ n,θ,x 1
) are bounded in L k -norm independently of n and x. From this, we deduce sup 0≤i≤n-1

n k-k/α E[|ξ i,n | k ] ≤ C(k), ∀k ≥ 1,
where the C(k) are some finite constants. It can be classically checked that the previous control, for instance with k = 4, is sufficient to imply the Lindeberg's condition (7).

Stable central limit theorem

This section is devoted to the proof of the stable convergence in law stated in Theorem 3.

Proof Since u n = n 1/2-1/α , we have

u n n-1 i=0 ξ θ i,n = n -1/2 n-1 i=0 n 1-1/α ṗθ 1 n (X θ i n , X θ i+1 n ) p θ 1 n (X θ i n , X θ i+1 n
) .

The Theorem 3 is an immediate consequence of the Lemmas 8-9 below.

Lemma 8

Set

η i,n = n 1-1/α ṗθ 1 n (X θ i n , X θ i+1 n ) p θ 1 n (X θ i n , X θ i+1 n ) -ḃ(X θ i n , θ) ϕ α (n 1/α (L i+1 n -L i n ) + κ n ) ϕ α (n 1/α (L i+1 n -L i n ) + κ n ) , then we have n -1/2 n-1 i=0 η i,n n→∞ ---→ 0.
Using the notations of Section 4, we define d n,θ,x 0 = E n 1-1/α qn,θ,x 0 (Y n,θ,x 0 1 ) q n,θ,x 0 (Y n,θ,x 0 1

) ḃ(x 0 , θ) ϕ α (L n,α 1 +κn) ϕα(L n,α 1 +κn) , so that the left hand side of (87) reduces, from the Markov property, to d n,θ,X θ i/n . From (37) and κ n → 0,

we have Y n,θ,x 0 1 -L n,α 1 + κ n ∞ n→∞ ---→ 0.
Using the fact that ϕ α ϕα has a bounded derivative (see e.g. Theorem 7.3.2 in [START_REF] Vassili | Markov processes, semigroups and generators[END_REF]) together with Cauchy-Schwartz inequality and Proposition 7 ii), we can deduce that,

d n,θ,x 0 = E n 1-1/α qn,θ,x 0 (Y n,θ,x 0 1 ) q n,θ,x 0 (Y n,θ,x 0 1 ) ḃ(x 0 , θ) ϕ α (Y n,θ,x 0 1 ) ϕ α (Y n,θ,x 0 1 ) + o(1),
where the o(1) term is uniform with respect to x 0 . Using the Proposition 4, we have

d n,θ,x 0 = E n 1-1/α H n θ ( Ẏ n,θ,x 0 1 ) ḃ(x 0 , θ) ϕ α (Y n,θ,x 0 1 ) ϕ α (Y n,θ,x 0 1 ) + o(1).
From the convergence result (74), we deduce that sup

x 0 d n,θ,x 0 -ḃ(x 0 , θ) 2 E H L α (1) ϕ α (Y n,θ,x 0 1 ) ϕ α (Y n,θ,x 0 1 ) n→∞ ---→ 0.
From Lemma 3 and (37), we can deduce that,

d n,θ,x 0 n→∞ ---→ ḃ(x 0 , θ) 2 E H L α (1) ϕ α (L α 1 ) ϕ α (L α 1 )
, uniformly with respect to x 0 . Then, the relation (78) enables to rewrite this convergence as,

d n,θ,x 0 n→∞ ---→ ḃ(x 0 , θ) 2 E ϕ α (L α 1 ) 2 ϕ α (L α 1 )
2 , uniformly with respect to x 0 .

This result implies (87) and hence (85).

Lemma 9 On has the convergence in law,

n -1/2 n-1 i=0 ϕ α (n 1/α (L i+1 n -L i n ) + κ n ) ϕ α (n 1/α (L i+1 n -L i n ) + κ n ) ḃ(X i n , θ) n→∞ ---→ N (0, I θ ), ( 88
)
where the convergence is stable with respect to G 1 .

Lemma 11 Let ξ : R → R be a bounded function with support included in [-1, 1], with bounded derivative and such that ξ ∈ L 1 ( dz |z| 1+α ) and let

I n (t) = (E n t ) p t 0 R (E n s ) -p ξ(z)µ(ds, dz), I(t) = t 0 R ξ(z)µ(ds, dz),
where p ≥ 1 is some real constant. Then, the following convergences hold in L q -norm for all q ≥ 1, -1)dt dz |z| 1+α , which is finite and bounded independently of n. This shows that the exponential moments of I are bounded. We deduce that R n (1) → 0 in L p norm, uniformly with respect to the parameter θ, and the lemma follows.

I n (1

Lemma 12

We have

Γ n 1-1/α Ẏ n,θ,x 0 1 , n 1-1/α Ẏ n,θ,x 0 1 Γ Y n,θ,x 0 1 , Y n,θ,x 0 1 ≤ Cn 1-2/α ,
where C is some constant independent of n, θ, x 0 .

Proof The process Ẏ n,θ,x 0 is solution of From this, we can deduce that Q t = Γ( Ẏ n,θ,x 0 t , Ẏ n,θ,x 0 t

) is solution of the equation,

Q t = t 0 2n -1 Q s b (x 0 +n -1/α Y n,θ,x 0 s , θ)ds+2n -1-1/α t 0
Γ(Y n,θ,x 0 that we will not fully use the explicit expression of H L α (1) in the proof. For the sake of shortness, let us denote H K = H L α (1)χ K (H L α (1)). The crucial facts about H K is that H K ∞ ≤ K and that it is a smooth Malliavin functional, with

Γ(H K , H K ) ≤ c 2 K Γ(H L α (1), H L α (1)) is element of p≥1 L p ,
where c K is any upper bound of the derivative of x → xχ K (x).

We now prove (98). Let us denote by H any primitive function of h. We compute the following expectation using the Integration by Part Formula (19) in Proposition 2,

E[h(L n,α 1 + κ n )H K ] = E H(L n,α 1 + κ n )H L n,α 1 (H K ) ( 99
)
where H L n,α 1 (H K ) is given by (20). Using the definition of Γ in [START_REF] Le | Asymptotics in statistics[END_REF], we get the following expression for the Malliavin weight

H L n,α 1 (H K ) = L( H K Γ(L n,α 1 , L n,α 1 )
)L n,α 1 -

L(L n,α 1 )H K Γ(L n,α 1 , L n,α 1 ) -L( H K L n,α 1 Γ(L n,α 1 , L n,α 1 )
) By (37) we have L n,α 1 -κ n -Y n,θ,x 0 1 ≤ b ∞ n 1/α-1 + κ n n→∞ ---→ 0, hence using that the function H is globally Lipschitz with a constant h ∞ , we deduce from (99) that

E[H K h(L n,α 1 + κ n )] -E H(Y n,θ,x 0 1 )H L n,α 1 (H K ) ≤ ε n h ∞ E H L n,α 1 (H K ) (100)
where (ε n ) n is some sequence converging to zero. We now compute E H(Y n,θ,x 0 1 )H L n,α 1 (H K ) using successively the self-adjoint property of the operator L and the chain rule, to obtain an I.P.P. formula in a reverse direction.

E H(Y n,θ,x 0 1 )H L n,α 1 (H K ) = E H(Y n,θ,x 0 1 ) L( H K Γ(L n,α 1 , L n,α 1
) ))L n,α 1 }H K Γ(L n,α 1 , L n,α 1 )

)L n,α 1 - L(L n,α 1 )H K Γ(L n,α 1 , L n,α 1 ) -L( H K L n,α 1 Γ(L n,α 1 , L n,α 1
= E H K Γ(L n,α 1 , H(Y n,θ,x 0

1

)) Γ(L n,α 1 , L n,α 1 )

= E H K h(Y n,θ,x 0 1 ) Γ(L n,α 1 , Y n,θ,x 0 1 ) Γ(L n,α 1 , L n,α 1 )

(101)

where α ∈ ( 1 , 2 )

 12 and τ is a non negative smooth function equal to 1 on [-1, 1], vanishing on [-2, 2] c and such that 0 ≤ τ ≤ 1. The introduction of the truncation function τ in the density of the Lévy measure ensures the integrability of |L t | p , ∀p ≥ 1.

Corollary 1 1 2 - 1 α

 111 The family (P θ n ) satisfies the LAMN property with rate u n = n , and information I θ

  n (1) -I(1), I n (1) -I(1)) n→∞ ---→ 0.Proof The convergence of I n (1) to I(1) is clear since s → E n s converges uniformly to the constant 1, and is bounded by above and below (recall (41)).We now focus on bracket Γ(I n (1) -I(1), I n (1) -I(1)). Let us remark that (I n (t)) t is solution to the linear equation,I n (t) = p t 0 I n (s)n -1 b (x 0 + n -1/α Y n,θ,x 0 s , θ)ds + I(t).

0 ( 1 , 1 ) 1 , Y n,θ,x 0 1 ) 1 , Y n,θ,x 0 1 ) 1 , Y n,θ,x 0 1 )

 011111111 We set W n (t) = I n (t)-I(t) and R n (t) = Γ(W n (t), W n (t)). The process W n satisfies the linear equationW n (t) = p t 0 n -1 [I(s) + W n (s)]b (x 0 + n -1/α Y n,θ,x 0 s , θ)ds.In turns, it can be seen that the processR n (t) is solution to R n (t) = pn -1/α-1 t 0 2(I(s) + W n (s))b (x 0 + n -1/α Y n,θ,x 0 s , θ)Γ(Y n,θ,x 0 s , W n (s))ds+ n -1 p t 0 2b (x 0 + n -1/α Y n,θ,x 0 s , θ)[Γ(I(s), W n (s)) + R n (s)]ds Using that Γ(Y n,θ,x 0 s , W n (s)) ≤ Γ(Y n,θ,x 0 s , Y n,θ,x 0 s ) 1/2 Γ(W n (s), W n (s)) 1/2 ≤ Γ(Y n,θ,x 0 s , Y n,θ,x 0 s )+Γ(W n (s), W n (s))and a similar control for |Γ(I(s), W n (s))| we get,R n (t) ≤ Cn -1/α-1 t |I(s)|+|W n (s)|)(Γ(Y n,θ,x 0 s , Y n,θ,x 0 s )+R n (s))ds+Cn -1 t 0 [Γ(I(s), I(s))+R n (s)]ds, where C is some constant depending on b ∞ , b ∞ Now, we recall the control Γ(Y n,θ,x 0 s , Y n,θ,x 0 s ) ≤CΓ(Y n,θ,x 0 Y n,θ,x 0 , for s ≤ 1, and use the controls|I(s)| + |W n (s)| ≤ C 1 0 R |ξ(z)| µ(ds, dz) := I Γ(I(s), I(s)) ≤ 1 0 R ξ (z) 2 ρ(z)µ(ds, dz) := J .We deduce,R n (t) ≤ C t 0 [n -1/α-1 I + n -1 ]R n (s)ds + Ct[n -1/α-1 I Γ(Y n,θ,x 0 + n -1 J ].We deduce thatR n (1) ≤ n -1 C exp Cn -1/α I + C (n -1/α I Γ(Y n,θ,x 0 + J ).Now, J and sup θ Γ(Y n,θ,x 0 have finite moments of any order, bounded independently of n.And using the exponential formula for Poisson measure, we have E exp Cn -1 1 0 R |ξ(z)| µ(ds,

Ẏ n,θ,x 0 t= n - 1 t 0 b

 010 (x 0 + n -1/α Y n,θ,x 0 s , θ) Ẏ n,θ,x 0 s ds + n 1/α-1 t 0 ḃ(x 0 + n -1/α Y n,θ,x 0 s , θ)ds.

1 )L n,α 1 ) 1 )L(L n,α 1 )

 1111 -H(Y n,θ,x 0 -L(H(Y n,θ,x 0 1

We turn now to the study of the derivative of q θ with respect to the parameter θ. By iterating the integration by part formula, since Y θ

This research benefited by the support of the '

Proof Using Lemma 9 in [START_REF] Genon | On the estimation of the diffusion coefficient for multidimensional diffusion processes[END_REF], it is sufficient to show :

We start by the proof of (84). Since a score function has an expectation equal to zero, and

is independent of G i/n , we deduce that

But, since (L t ) t has stationary increments, the law of

) is the same as the law of L n,α 1 . We know from Lemma 3, that P (L n,α

where we used that ϕ α ϕα is bounded (see e.g. Theorem 7.3.2 in [START_REF] Vassili | Markov processes, semigroups and generators[END_REF]). Using E ϕ α (L α 1 ) ϕα(L α 1 ) = R ϕ α (u)du = 0, we deduce [η i,n | G i/n ] ≤ Cn -1 for some constant C and (84) follows.

We now prove (85). Recalling the definition [START_REF] Jeganathan | Some asymptotic properties of risk functions when the limit of the experiment is mixed normal[END_REF], we have

With a method analogous to the proof of (84), we can show that E

. From Proposition 7, it appears that the first two terms in the Right Hand Side of (86) are asymptotically closed to the same quantities, and that (84) is proved as soon as we show the following control holds, uniformly with respect to i,

Proof Let us define the processes,

, θ),

We will apply Lemma 2.8 in [START_REF] Jacod | The Euler scheme for Lévy driven stochastic differential equations: limit theorems[END_REF] to prove (88). Indeed, we will show that there exists a Gaussian random variable γ, independent of L 1 , such that one has the convergence

Then, by application of Lemma 2.8 in [START_REF] Jacod | The Euler scheme for Lévy driven stochastic differential equations: limit theorems[END_REF], there exists a Brownian motion (Γ t ) t independent of (L t ) t such that one has the convergence in law for processes

This exactly implies the lemma, if we show furthermore var(Γ 1 ) = var(γ) = E ϕ α ϕα (L α 1 ) 2 . Let us focus on the derivation of the convergence (89). For (u, v) ∈ R 2 , let us set

Using the i.i.d. structure of the increments of the Levy process L, we easily get the following expression

Let us study the asymptotic behaviour of χ n (u, v). Using that ϕ α /ϕ α is bounded we get

First, we have

where ψ(v) is the Lévy Kintchine exponent of L 1 .

We now focus on the term χ

n (u, v). Using ( 35)-(36) of Lemma 3, and the fact that n 1/α L 1/n has the same law as L n,α 1 , we get

) using integration by part formula

For the term, χ

n (u, v) using Lemma 3 again, it is easy see that

Collecting together (90)-( 95), we have

and thus the convergence (89) with γ ∼ N 0, E ϕ α ϕα (L α 1 ) 2 .

Appendix

Proof of Lemma 10

We prove in this section the following result.

Lemma 10

We have for all p ≥ 1,

ḃ(x 0 , θ)H (2) , where H (2) is a random variable that can be expressed as a functional of the random measure µ and the function ρ.

Proof We first show two intermediate results that are useful for the proof of the lemma.

The lemma is proved.

We are now able to prove the Lemma 10.

From the definition (20) and the basic properties of (•,

We deduce that, for any p ≥ 1,

Now, the explicit expression for Ẏ n,θn,x 0

Thus, we need to prove the convergence of H Y n,θn,x 0 1 ( H n θn (1)) to H (2) , depending only on ρ and µ. Recalling (20) and using basic properties of the operator Γ, we have

The convergence of the first two terms of the Right Hand Side of this equation follows from computations analogous to the proof of Proposition 5. Indeed, in the proof of Proposition 5, it is

shown that H n θn (1) converges to H L α (1) defined by (54). The convergence of Γ(Y n,θn,x 0 1 , Y n,θn,x 0

1

),

)) and L(Y n,θn,x 0

1

) to quantities independent of b can be obtained by studying their respective explicit expressions (43)-(45).

It remains to study the convergence of Γ(Y n,θn,x 0 1 , H n θn (1)). After cumbersome computations relying on (40), (54), the fact that As a consequence, we have Γ(Y n,θn,x 0

. Using (37), we deduce

where at the last line we have used that the quantity Γ(Y n,θn,x 0 s , Y n,θn,x 0 s

) is bounded in L p norm and that Γ(H L α (1), H L α (1)) has finite moments of any order.

Recalling the expressions (33), (54) and using the basic properties of the operator Γ, one can see that the computation Γ(L n,α 1 , H L α (1)) can be reduced to the computation of the Γ-bracket between simple stochastic integrals. Moreover, since ρ is supported on [-1, 1], such computations show that

). This ends the proof of the lemma.

Regularity of the conditional expectation

Let us recall that we have defined the functions η n and η by the relations

The aim of the section is to show that the function η n and η are close in some sense.

We recall that κ n is defined in Lemma 3. Our first result is the following.

Proposition 9 There exists a sequence (ε n ) n , independent of x 0 and θ, with ε n → 0, such that the following holds true. For all h bounded smooth function,

Proof Remark that (97) is a result about the total variation distance between the laws of Y n,θ,x 0 1 and L n,α 1 + κ n , and (98) will be useful to control difference between the conditional expectations of

). We shall only prove (98) since (97) can be obtained in a similar way. Remark

Putting together (100) and (101) we deduce

-1

1

Hence the proposition will be proved if we show

-

To prove (102), we write from (20)

.

From the fact that H K is bounded and Γ(H K , H K ) admits finite moments, together with the fact that Γ(L n,α 1 , L n,α 1 )), Γ(L n,α 1 , Γ(L n,α 1 , L n,α 1 )) and L(L n,α 1 ) do not depend on n (this is due to the choice of the support of ρ), we easily see that H L n,α 1 (H K ) admits moments bounded independently of n.

To prove (103), we write by (37),

we easily derive

and thus (103).

Corollary 2 There exists a sequence (ε n ) n , independent of x 0 and θ, with ε n → 0, such that the following holds true. For all h bounded smooth function,

Proof From Lemma 3, we know that P (L n,α 1 + κ n = L α 1 )

n→∞ ---→ 1. Hence we deduce (104) from (97).

Using that

) is bounded we deduce (105) from (98).

Proposition 10 We have

))

and this convergence is uniform with respect to x 0 , θ.

Proof We estimate the L 1 norm appearing in (106) by duality. Let β : R → [-1, 1] be a measurable function, we evaluate :

) -η(Y n,θ,x 0


Aymen Mudheher Badr
email: aymen.m.badr@uodiyala.edu.iq

Lamia Chaari Fourati

Samiha Ayed
email: samiha.ayed@utt.fr

An Improved GIFT Lightweight Encryption Algorithm to Protect Medical Data In IoT

Keywords: IoT, Chaotic, Security, GIFT, IoMT

The Internet of Things (IoT) enables the interconnection of devices that collect massive amounts of data, making IoT security requirements crucial and secured through encryption. However, traditional encryption protocols are no longer suitable for all IoT scenarios. We propose a new lightweight encryption method optimized for patient information protection in healthcare, considering the limited capacity of portable medical devices. Our method has been experimentally demonstrated to be effective, with a largest entropy of 7.9917 in medical images (computed tomography) and average coding and decoding times of 3.8952 sec and 3.0584 sec, respectively. The encoded image exhibits an even distribution of pixels and lower correlation coefficients between neighboring pixels, supporting the effectiveness of our less complex method compared to current state-of-the-art methods.

I. INTRODUCTION

For Internet of Things (IoT) devices that are limited in storage space, processing speed, or battery life, a specialized field of cryptography known as lightweight encryption has been developed. Many commonly used encryption methods are vulnerable to attack. Lightweight Encryption Algorithm (LEA) is just one of many such algorithms available, including several others with varying degrees of complexity. Consequently, there is a rising need to develop and implement lightweight encryption algorithms to solve the challenge of applying encryption methods to resource-restricted devices on the Internet of Things setting. Using an encryption key or keys, cryptography is the practice of making data or messages unreadable to the naked eye [START_REF] Kubba | Developing a lightweight cryptographic algorithm based on DNA computing[END_REF]. Over time, several different encryption algorithms have been developed, each with its own set of pros and cons. These algorithms are time-consuming to implement, energy-intensive, and resource-intensive. A newer technique dubbed a "lightweight" encryption system has been introduced due to the need for a different approach to encryption for low-powered devices (such as portable medical equipment and IoT gadgets) [START_REF] Shetty | A survey on performance analysis of block cipher algorithms[END_REF].

Substitution permutation networks (SPNs), represented by the structure of repeated product zeros, were proposed by Feistel in 1975 and are among the most well-known and ancient scientific mathematical sequences used in the cypher area for blocks. S-boxes, also known as substitution permutation networks, are used to link together a sequence of substitution rounds [START_REF] Sehrawat | Lightweight block ciphers for IoT based applications: a review[END_REF]. The GIFT cryptographic system, which combines a block cipher algorithm with a substitution permutations network example, is one of the simplest (and most effective) [START_REF] Heys | A Tutorial on the Implementation of Block Ciphers: Software and Hardware Applications[END_REF]. The GIFT algorithm's implementation mechanism consumes little resources and produces accurate results while being extremely lightweight and memory-efficient. Since the algorithm's (Add Round Key) feature exploits the algorithm's weak key mechanism, it is vulnerable to information leakage [START_REF] Cohney | Pseudorandom black swans: Cache attacks on CTR-DRBG[END_REF].

In encryption, a "key" is a piece of data used to encrypt and decrypt files. It can be a string of numbers or letters that, when run through an algorithm, reveals the encrypted or decrypted data. The strength of encryption depends on the size and nature of the key used. It's essential to keep the key secret to maintain encryption integrity. Several factors, such as algorithm, key size, generation, and exchange, contribute to overall security. Randomly generated keys with enough entropy to avoid guessing are necessary [START_REF] Mousavi | Security of internet of things based on cryptographic algorithms: a survey[END_REF]. Random key generation is a complex issue and has been addressed in various ways by cryptographic systems [START_REF] Salau | Data security in wireless sensor networks: attacks and countermeasures[END_REF].

The output of a machine called a Random Bit Generator (RBG) can be used to produce a key directly. RBGs produce a sequence of bits that are completely random and unbiased. To create an asymmetric key pair, RBG can be used to generate either a symmetric key or random data. Alternatively, a key can be produced indirectly from another key or a password during a key agreement process [START_REF] Dasgupta | A survey of blockchain from security perspective[END_REF].

Mathematical methods in chaotic cryptography use chaos theory to securely transmit information between parties or adversaries. Despite interest in this approach since 1989, concerns around security and speed have slowed its deployment [START_REF] Pandey | Implementing healthcare services on a large scale: challenges and remedies based on blockchain technology[END_REF]. A chaotic cipher consists of two halves: the cipher itself and the analysis of the cipher. Cryptanalysis involves deciphering encrypted messages, while cryptography involves encoding information for secure transmission [START_REF] Anwar | A pixel permutation based image encryption technique using chaotic map[END_REF]. Effective implementation of chaos theory in coding requires chaotic maps, which use the map's entropy to generate necessary noise and propagation. This research proposes a hybrid method to generate cryptographic keys, using a pseudo-random number generator based on a chaotic system type (Lozen's approach) or Chaos Theory, to improve the performance of the GIFT algorithm [START_REF] Lee | Blockchain-based healthcare information preservation using extended chaotic maps for HIPAA privacy/security regulations[END_REF].

Encryption is a crucial data protection method used in healthcare to safeguard electronic health records (EHR) and personal health information (PHI) against malicious intent and privacy breaches. Portable medical devices used to transmit patient data over the Internet of Things require protective techniques that do not drain power reserves. This work proposes a lightweight encryption algorithm to protect medical files and implements blockchain to authenticate data authenticity during storage. The proposed algorithm produced favorable results when compared to older encryption methods.

The following is the outline that this paper will adhere to: references to related literature are provided in Section II. In this section, the suggested methodology is broken down into its most fundamental components and laid forth for consideration in Section III. Section V provides a thorough explanation of the proposed systems used in Section IV. Metrics of performance are explained in Section V. The proposed system's effectiveness is evaluated and explained in Section VI. The concluding findings of the study are presented in Section VII of the article.

II. RELATED WORKS

In this section, the significant literature that has been published in recent times will be discussed. The researchers begin with a summary of prior work in this area and a discussion of its main shortcomings when it comes to protecting the confidentiality of patient's medical records. The proposed approach is briefly described at the end of this section. Privacy of patients' medical information is important because it is becoming a valued commodity. Compression, Authentication, Hybrid, and Encryption are just a few of the cryptographic techniques presented to ensure the safety of videos and digital images. Encryption strategies based on the most recent findings have been recommended to protect users' personal information [START_REF] Wu | Image encryption using the two-dimensional logistic chaotic map[END_REF], [START_REF] Belazi | A novel image encryption scheme based on substitution-permutation network and chaos[END_REF].

For instance, Wei et al. [START_REF] Wei | Autoprivacy: automatic privacy protection and tagging suggestion for mobile social photo[END_REF] A new encryption cryptosystem has been developed for multimedia social networks, using transformation techniques like the Latin square, chaotic maps, neural networks, and DNA technique encryption. Chaotic maps encryption is particularly efficient and widely used in contemporary cryptography. The system allows approved users to obscure unwanted human faces by restricting blurring capabilities.

Nonlinear dynamics are presented in two forms in the literature: discrete systems, like a logistic map, and continuous systems, like a hyper-chaotic system [START_REF] Wei | Trustworthy authentication on scalable surveillance video with background model support[END_REF]. These systems' primary cryptographic utility is their ability to produce random number sequences with high-quality stochastic behavior. Recently, Hamza et al. [START_REF] Hamza | A novel pseudo random sequence generator for imagecryptographic applications[END_REF] presented a Zaslavsky chaotic map-based encryption cryptosystem for digital image security. Without any finite-precision computing, the resultant chaotic signals were employed immediately in the permutation stage.

Furthermore, the starting values of the chaotic maps are quite sensitive to change. As a result, these seed values served as the private keys in chaos-based encryption systems developed by cryptographers. Pseudo-random number generator (PRNG) algorithms based on chaotic systems are well-known examples of such cryptographic applications. The commonly-employed pseudo-random number generator for chaotic-encryption-scheme stream-key generation. To encrypt images and other data, one example is Hu et al. [START_REF] Hu | Pseudorandom sequence generator based on the chen chaotic system[END_REF] the presentation of a pseudo-random number generator (PRNG) based on a high-dimensional chaotic map, which makes use of the combination of three coordinates of the chaotic orbits.

Furthermore, Hamza et al. [START_REF] Hamza | A novel sensitive image encryption algorithm based on the Zaslavsky chaotic map[END_REF] suggest a patient-privacyprotecting chaos-based encryption cryptosystem. The suggested cryptosystem protects patient photos from hacked brokers. For the purpose of prioritizing and securing medical keyframes extracted during a wireless capsule endoscopy surgery, we present a rapid probabilistic cryptosystem. Our cryptosystem's encrypted images are random, ensuring computational efficiency and keyframe security against multiple assaults. Additionally, it decrypts medical data only for authorized users to protect patient privacy [START_REF] Hamza | A privacy-preserving cryptosystem for IoT Ehealthcare,InformationSciences[END_REF].

Hasan et al. [START_REF] Hasan | Lightweight Encryption Technique to Enhance Medical Image Security on Internet of Medical Things Applications[END_REF] provides a fast, lightweight encryption algorithm for healthcare image encryption. The lightweight encryption method secures medical images using two permutation methods. Security and execution time are compared to normally encrypted methods. The algorithm was tested on many photos. Masood et al. [START_REF] Masood | A Lightweight Chaos-Based Medical Image Encryption Scheme Using Random Shuffling and XOR Operations[END_REF] provide a fast, lightweight encryption algorithm for healthcare image encryption. The lightweight encryption method secures medical images using two permutation methods. Security and execution time are compared to normally encrypted methods. Numerous test photos have been utilized to evaluate the proposed technique.

III. THEORETICAL BACKGROUND

A. Internet of Medical Things

New technology and energy sources are introduced every day, including the latest 1G to 5G [START_REF] Fadhil | Evolutionary perspective of mobile communication technologies[END_REF] networks used in IoT applications and systems. However, high bandwidth and frequency may pose privacy and security concerns, requiring proper protocols and policies to address them. Medical devices and systems, such as consumer health monitors and pacemakers, communicate wirelessly via Bluetooth or proprietary protocols, transmitting vital data to doctors and patients. However, legacy medical devices and techniques still in use may not support modern Endpoint Detection and Response (EDR) systems, leading to security and privacy concerns. To address these issues, three factors must be considered for IoMTenabled security: identifying networked devices, enforcing communication protocols, and checking for tampering with internal systems. Encryption is a useful tool for safeguarding diagnostic images in IoMT applications, where unauthorized image copying or usage may occur [START_REF] Rougaii | VIRTUAL DOCTOR SYSTEM BASED ON H-IOT APPLICATIONS OVER 5G NETWORK: A COM-PARISON STUDY[END_REF].

B. GIFT Block Cipher

The GIFT block cipher uses symmetric key cryptography with the Substitution Permutation Network (SPN) method. Both GIFT-64/128 (a 64-bit block with a 128-bit key) and GIFT-128/128 are viable choices (128-bit block and 128-bit key). The four operations of Sbox, Permutation, AddRound-Key, and Constant XOR are performed in each round of the GIFT block cipher. Figure 1 depicts the encipherment procedure of the GIFT block cipher [START_REF] Jang | Grover on GIFT[END_REF].

1) GIFT Block Cipher Sbox:

The 4-bit Sbox's input value is the n-bit block (n = 64, 128) after it has been divided into 4 bits. Table 1 provides the GIFT block cipher's Sbox [START_REF] Hamza | A novel sensitive image encryption algorithm based on the Zaslavsky chaotic map[END_REF].

2) GIFT Block Cipher Permutation:

The GIFT-64/128 permutation swaps out the P64 i-th bit of block B with the i-th bit of block B. Table 2 provides information on the permutation of GIFT-64/128. This document omits the full Table on GIFT-128/128 permutation [START_REF] Hamza | A novel sensitive image encryption algorithm based on the Zaslavsky chaotic map[END_REF].

3) GIFT Block Cipher's AddRoundkey: k0 and k1 (a combined 32-bit value) are chosen from the key (K =k7,..., k0) in the GIFT-64/128 block cipher. In the round key, k0 and k1 are utilized as U and V, respectively. RK = U V =u15...u0 v15...v0 (U = k1, V = k0). The round key is XORed to b4i+1, and V to b4i to form an exclusive-ore with the block B. In the GIFT-128/128 block cipher, k0, k1, k4, and k5 (64bit in a total) are selected from the key K. k0, k1, k4 and k5 are used as U and V of the round key as follows, RK = U V = u31...u0 v31...v0 (U = k5 k4, V = k1 k0). The round key is XORed to the block B, where U is XORed to b4i+2 and V is XORed to b4i+1. b4i+2 ← b4i+2⊕ui, b4i+1 ← b4i+1⊕vi, i = 0, ..., 31 (2)

b 4i+ 1 ← b 4i+ 1 ⊕ u i , b 4i ← b 4i ⊕ v i , i = 0, ..., 15 (1)

4) GIFT Block Cipher Constant XOR:

The GIFT-64/128 and GIFT-128/128 block ciphers employ the round constants C listed in Table 3. Block B is created by XORed singlebit and round constants (C = c5c4c3c2c1c0) together as in Equation ( 3).

5) GIFT Block Cipher Keyschedule:

The Key schedule in GIFT-64/128 and GIFT-128/128 block ciphers updates the key (K = k7,..., k0) and extracts the round key from the updated key K. Equation (4) depicts the Key schedule. The symbol (i) represents a right rotation operation (i-bit) [START_REF] Hamza | A novel sensitive image encryption algorithm based on the Zaslavsky chaotic map[END_REF].

k 7 k 6 ... k 1 k 0 ← k 1 ⟩⟩⟩2 k 0 12 ... k 3 k 2 , (4)

C. Chaotic Chen System

Chaos theory is a mathematical and scientific study of highly sensitive dynamical systems that were previously thought to be completely chaotic and irregular. Even seemingly chaotic and complex systems exhibit underlying patterns, repetition, self-similarity, fractals, and self-organization to those knowledgeable in chaos theory. The butterfly effect as of Figure (2), a central notion of chaos theory, describes how a small perturbation in one state of a nonlinear system can have a significant impact on another state due to sensitive dependence on initial conditions. The proposed strategy utilizes the Lorenz Attractor, which demonstrates how data changes over time and shows that it is almost impossible to know all the infinite numbers of data [START_REF] Masood | A Lightweight Chaos-Based Medical Image Encryption Scheme Using Random Shuffling and XOR Operations[END_REF].

Edward Lorenz's 1963 atmosphere model simplification was a significant step forward in understanding atmospheric change, as it simplified the process to a differential equation in Eq.5.:

dx dt = α(y -x) (5)
Fig. 2. Lorenz attractor

dy dt = x(β -z) (6)
dz dt = xy -γz (7)
Where: Alpha is a Prandtl number, Beta is proportional to the Rayleigh number Gamma is a geometric factor.

IV. THE PROPOSED SYSTEM

This innovative way to generate random keys for chaotic systems is called Chaos Theory, and it is the basis for the suggested method (CSKey) (Lozens method). The CSKey produces a string of completely random digits. Using these random numbers and a lightweight GIFT algorithm, CSKey performance was enhanced and a robust chaotic encryption mechanism was made available for Internet-of-Things (IoT) equipment transmitting medical data. It is demonstrated in

V. THE METRICS OF PERFORMANCE

Performance quality is a cornerstone criterion for assessing lightweight encryption methods. Here, we'll detail our research into the GIFT algorithm's implementation and show how we tested it, both with and without the proposed Chaos Secret Key (CSKey). We will evaluate the effectiveness of the proposed key in the algorithm by comparing it to the standard approach. Performance metrics refer to a wide variety of indicators that have widespread worldwide recognition [START_REF] Fadhil | Evolutionary perspective of mobile communication technologies[END_REF], [START_REF] Rougaii | VIRTUAL DOCTOR SYSTEM BASED ON H-IOT APPLICATIONS OVER 5G NETWORK: A COM-PARISON STUDY[END_REF].

• Entropy, High-entropy images are more secure as they contain more randomness, making decryption without the

where: H is the entropy of the system, p(i) is the probability of the i-th possible outcome of the system, and log2 is the base-2 logarithm.

• Mean squared error (MSE) is the most widely used estimate statistic for image quality. as seen in Eq. [START_REF] Dasgupta | A survey of blockchain from security perspective[END_REF].

M SE = 1 n n i=1 [I(i) -k(i)] 2 (9)
• Peak Signal-to-Noise Ratio (PSNR) measures signal representation quality relative to noise distortion in Eq.( 9). It includes both extreme and intermediate values for quality assessment.

P SN R = 20. log 10 (M AX i ) -10. log 10 (M SE) [START_REF] Moin | Securing IoTs in distributed blockchain: Analysis, requirements and open issues[END_REF] where: MAXi is the maximum possible pixel value of the image.

• Measurement of the Universal Quality Image Index(UQI): measures image quality after modifications, is easy to compute, and can be applied to various image processing programs. [START_REF] Masood | A Lightweight Chaos-Based Medical Image Encryption Scheme Using Random Shuffling and XOR Operations[END_REF].

U QI = 1 1 M m j=1 Qj (11)
• Measuring the structural similarity index (SSIM), how similar a recovered image is to the original by calculating the structural similarity index based on an inferential model that perceives image degradation as a shift in structural information.

SSIM (x, y) = [l(x, y)] α * [c(x, y)] β * [s(x, y)] γ (12)
• SCC measures spatial correlation and is similar to measuring the intensity of a linear relationship between x and y, where the average gain of the incoming signal correlates with its angle of arrival.

SCC = [ (I(x, y) -µI) * (K(x, y) -µK)] [σI * σK] (13)
• Number of Changing Pixels Rate (NPCR) is the rate at which the cipher picture's pixel count shifts in response to a shift of a single pixel in the plain image.

N P CR = N c N * 100% (14)
where: Nc is the number of pixels that change between the two encrypted images, and N is the total number of pixels in the images.

• Unified Average Changing Intensity (UACI) is used to determine the level of dissimilarity between a plain image and a ciphered image.

U ACI = (1/n) * |I(x, y) -K(x, y)| M AXi ( 15
)
where: I(x, y) and K(x, y) are the intensities of the two images at the pixel location (x, y), n is the total number of pixels in the images, and MAXi is the maximum possible pixel value of the image (e.g., 255 for an 8-bit grayscale image).

VI. RESULT AND DISCUSSION

The researchers have conducted a variety of experiments to measure the robustness and reliability of the suggested cryptosystem. Experiments and measurements are taken to ensure the system is secure against several types of attacks, such as statistical, secret-key, and differential ones as shown in Table ( 4) and ( 5), and Table ( 6) and ( 7) shone the values of correlation.To begin, a histogram analysis is presented to show that the encrypted images' pixels are distributed uniformly and in a way that is fundamentally different from the histograms of the plain images. The next step is to calculate the image's information entropy, which proves the image's randomness and its resistance to entropy attacks. The correlation coefficient demonstrates that the suggested cryptosystem reduces the correlation between neighboring pixels of the plain image sequentially. Differential attacks are evaluated with the help of the number of pixels changing per second (NPCR) and the unified average changing intensity (UACI) tests. The power of our cryptosystem to withstand extensive attacks is uncovered through analysis of the secret keys. The probabilistic analysis carried out here demonstrates that the same keyframe and the same secret keys result in radically distinct encrypted keyframes produced by the cryptosystem. In addition, the sensitivity analysis proves beyond a reasonable doubt that recovering the original frames from the backup requires knowing the precise values of the secret keys. Thus, the encrypted image will change drastically if the secret keys are ever changed. Image quality analysis further substantiates the excellent state of the encrypted image. To ensure that a cryptosystem is secure, anti-clipping analysis simulates scenarios in which some pixels from some blocks are clipped off. In the end, the comparative study proves that our cryptosystem is far superior to the current state of the art.

Histograms plot the number of pixels at each level of color intensity, providing a visual representation of the image's pixel distribution. The goal of this check is to ensure that the histogram and distribution of pixel values in the encrypted image are different from the original. The original, encrypted, and decrypted photographs are all displayed in Figure [START_REF] Cohney | Pseudorandom black swans: Cache attacks on CTR-DRBG[END_REF]. Histograms of the keyframes and encrypted images for each of the three RGB components are displayed in Figure [START_REF] Mousavi | Security of internet of things based on cryptographic algorithms: a survey[END_REF]. Histograms of the three encrypted images are consistent and dissimilar to those of the keyframes. The suggested cryptosystem's efficacy in removing correlation between neighboring pixels of a keyframe is demonstrated by the correlation coefficient test. When two random variables are correlated, it shows how strongly and in what direction they are related along a linear axis. This allows for analyzing the relationship between neighboring pixels in both the encrypted and unencrypted versions of the keyframe. Given the vast number of possible pixel combinations in a medical keyframe, randomly select 1024 pairs of adjacent pixels to examine for correlation in the x, y, and z axes, as shown in Figure [START_REF] Salau | Data security in wireless sensor networks: attacks and countermeasures[END_REF]. The comparative test is an important demonstration of the proposed cryptosystem's overall efficiency in relation to other state-of-the-art cryptosystems, particularly in ensuring rising suitability for real-world applications. As was demonstrated in the preceding subsections, we carried out a variety of comparison experiments. The findings revealed that our cryptosystem possesses superior performance when measured against other published studies [14] - [START_REF] Wei | Trustworthy authentication on scalable surveillance video with background model support[END_REF], [START_REF] Hu | Pseudorandom sequence generator based on the chen chaotic system[END_REF], [START_REF] Masood | A Lightweight Chaos-Based Medical Image Encryption Scheme Using Random Shuffling and XOR Operations[END_REF]. Here, the results show how the proposed cryptosystem stacks up against other cutting-edge encryption solutions by conducting more extensive comparison tests based on a variety of experimental analyses. According to Table [START_REF] Mousavi | Security of internet of things based on cryptographic algorithms: a survey[END_REF], the suggested cryptosystem is just as effective as competing systems when measured against a variety of analysis metrics (NPCR and UACI). Taking into account these advantages, the suggested cryptosystem can ensure patients' confidentiality and safety. hospitals and clinics, sensitive medical data is protected by the suggested cryptosystem. This cryptosystem makes use of a variant of the GIFT algorithm that makes use of two different types of chaotic maps. The encryption keys for the proposed improved GIFT are generated by combining and cascading the orbits of two of the 2D chaotic maps. In this proposed cryptosystem, a block symmetric encryption technique was used, constructed from a single round of confusion and diffusion. According to the study, the present approaches provide key-based, non-systematic sequence numbers that require lengthy computations. Comparatively, it is clear from the outcome that the computation required by the suggested algorithm is little. As a result, the suggested algorithm is carefully constructed to obtain the highest level of security in order to secure the medical image. The image is encrypted using three steps of encryption using a 256-bit key value for logical operation. The suggested cryptosystem also utilized the keyframe boundary to secretly incorporate noise (random) values that would not be discernible in the decoded image. The performance of our cryptosystem is exceptional, and it is able to effectively withstand a wide variety of assaults, including differential, statistical, and exhaustive attacks for the purpose of discovering secret keys. Extensive experimental findings and security analysis showed that the suggested cryptosystem is both quicker and safer than existing methods. When using the proposed cryptosystem, sensitive medical data contained in keyframes is protected. In addition to protecting patients' confidentiality, this method also saves time, money, and resources by decreasing the need for power, communication, analysis, and searching. To further increase the effectiveness of the proposed cryptosystem and to open new directions in this field of research, the researcher plans to investigate access control mechanisms and homomorphic encryption algorithms in the future.

Fig. 1 .

 1 Fig. 1. Encryption process of GIFT block cipher.

b

  n1 ← bn 1 ⊕ 1, b 23 ← b 23 ⊕ c 5 , b 19 b 1 9⊕c 4 b 15 b 15 ⊕c 3 , b 11 ← b 11 ⊕c 2 , b 7 7⊕c 1 , b 3 b 3 ⊕c 0 .(3)

Figure ( 3 )

 3 and Figure (4) that the suggested technique uses the initial values for these maps.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Original Encryption algorithm (Gift)

Fig. 5 .

 5 Fig. 5. Photographic encryption tests (a) Original Images (b) Encrypted Images and (c) Decrypted Images

Fig. 6 .

 6 Fig. 6. The histogram test of the Original Image and encrypted image with GIFT in the three Red, Green, and Blue components

Fig. 7 .

 7 Fig. 7. Before encryption, horizontal, vertical, and diagonal correlation of two neighboring pixels in the three spectral components (R, G, and B)

TABLE I SBOX

 I OF GIFT BLOCK CIPHER

TABLE II GIFT

 II -64 BIT PERMUTATION i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P 64 (i) 0 17 34 51 48 1 18 35 23 49 2 19 16 33 50 3 i 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 P 64 (i) 4 21 38 55 52 5 22 39 36 53 6 23 20 37 54 7 i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 P 64 (i) 8 25 42 59 56 9 26 43 40 57 10 27 24 41 58 11 i 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 P 64 (i) 12 29 46 63 60 13 30 47 44 61 14 31 28 45 62 15

TABLE VI CORRELATION

 VI COEFFICIENTS BETWEEN ADJACENT PIXELS TO ORIGINAL GIF

	Correlation	Correlation	Correlation
	Horizontal	Vertical	Diagonal
	-0.0054	0.01807	0.0498
	0.0112	-0.0144	0.0023
	0.0324	-0.0171	-0.0024
	0.0135	0.0305	0.0057
	-0.0193	-0.0059	-0.0101
	-0.0155	0.0024	-0.0081

TABLE VII

 VII

	CORRELATION COEFFICIENTS BETWEEN ADJACENT PIXELS TO MODIFIED
		GIF
	Correlation	Correlation	Correlation
	Horizontal	Vertical	Diagonal
	-0.0047	0.0012	0.0204
	0.0016	0.0168	-0.0234
	-0.0074	-0.0057	0.0096
	0.0731	0.0183	-0.0016
	-0.0122	-0.0231	0.0020
	0.0117	-0.0075	-0.0206

TABLE VIII ANALYSES

 VIII BASED ON A NUMBER OF DIFFERENT METRIC COMPARISONS

	Method	Image size	NPCR	UACI	Sensitivity
			analysis	analysis	analysis
	Proposed	[256,256,3]	99.6231	31.2235	Yes
	[23]	[640,480,3]	99.609	33.465	Yes
	[16]	[1024,1024,1] 99.617	33.669	Yes
	[19]	[256,256,1]	99.61	33.50	Yes
	[17]	[256,256,3]	99.217	33.405	Yes
	[14]	[640,480,3]	99619	33.477	Yes
	[15]	[640,480,3]	99.606	44.486	Yes
	VII. CONCLUSION AND FUTURE WORK
	This paper suggested a fast and reliable picture cryptosys-
	tem to keep medical records private. During its transfer to