# Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Authors: Wissem Jedidi
Journal: Complexity
 (2021)

---

## Abstract

We consider statistical experiments associated with a Lévy processX=Xtt≥0 observed along a deterministic scheme iun,1≤i≤n. We assume that under a probability ℙθ, the r.v. Xt,t>0, has a probability density function >o, which is regular enough relative to a parameter θ∈0,∞. We prove that the sequence of the associated statistical models has the LAN property at each θ, and we investigate the case when X is the product of an unknown parameter θ by another Lévy process Y with known characteristics. We illustrate the last results by the case where Y is attracted by a stable process.


---

## Body

## 1. Introduction

This work is a part of an ambitious program consisting in the estimation of the parameterθ intervening in the stochastic differential equation driven by a known Lévy process Y:(1)dXt=bθ,Xdt+aθ,XdYt,These kinds of models are motivated by mathematical finance problems ([1]). In this context, the property of local asymptotic normality property (LAN) has become an important issue [2]. The LAN property is described as follows: a sequence of families of probabilities ℙθnθ∈Θ indexed by an open set Θ⊂ℝ is said to have the LAN property at each point θ0∈Θ with speed n, if the sequence of probabilities localized around θ0,(2)ℙθ0+n−1/2θnθ∈ξ/θ0+n−1/2ξ∈Θ,converges, in the sense of weak convergence of the associated likelihood processes, to a Gaussian shift ℙθ′θ∈ℝ; see Section 2 for a precise definition. The LAN property allows to recover the so-called asymptotic Fisher information quantity Iθ0. This quantity is crucial in any estimation procedure, since 1/Iθ0 provides the lower bound of the variance of any estimator of θ0.The LAN property was investigated by Akritas [3] in models associated with Lévy processes X observed continuously in time over the interval 0,n,n⟶∞. He obtained the property under the assumption of differentiability, according to the parameter θ, of the characteristics bθ,cθ,νθ of X. With the same asymptotic, Luschgy [4] obtained the local asymptotic mixed normality (LAMN) property on models associated with semimartingales. As a notion, LAMN property is more general than the LAN because it allows the Fisher information quantity to be random. With the asymptotic 0,n,n⟶∞, the estimation methods do not seem to be feasible in practice, for this reason, several recent works focused on discretized schemes, i.e., observations of the process X along the discrete scheme(3)Xiun,1≤i≤n,n⟶∞.In practice, the most interesting case of the discretization pathun=1/n turns out to be relatively difficult. The classical case of a Brownian motion Y in (1) has been widely treated [5]. Clément and Gloter [6] studied the LAN for the model in (1), in the case where Y is a Lévy process attracted by a symmetric stable process with index α∈1,2. Aït-Sahalia and Jacod [7], Masuda [8], and Kawai and Masuda [9, 10] studied LAN property for the model in (1) in case of constant coefficients, i.e.,(4)dXt=θ1dt+θ2dYt,Our investigation goes to same direction of Aït-Sahalia and Jacod [11], who studied the LAN property and the problem of estimation of the parameter θ1,θ2 involved in the model of a log-asset price X, solution of (3) with Y being a standard symmetric stable process with index α∈0,2. Section 4 completes their situation in case where Y is a general stable process, eventually mixed. The last direction was initiated Rammeh [12] with observations according to random schemes Ti,n,1≤i≤n for the scale model:(5)X=θY,where θ is a real unknown real parameter, and Y is a symmetrical standard α-stable process. Rammeh showed that the LAN property always occurs, and his main arguments strongly rely to the linearity in θ to the fact that stable processes have the temporal scaling property and to the asymptotic behavior of the stable densities. Theorem 2 generalizes Rammeh’s results in the context of deterministic discrete scheme Ti,n=iun.Because of the intricacy of the case (1), we first focus on the following model, which contains (5) and intercepts (1): we assume that for all θ∈Θ, under ℙθ, X is a Lévy process, null at t=0, such that its Lévy exponent is given by the so-called Lévy-Khintchine formula:(6)EℙθeiuXt=etφθu,φθu=iubθ−cθ2u22+∫ℝeiuy−1−iuy1ly≤1μθdy,where bθ∈ℝ,cθ∈ℝ+, and μθ is a positive measure on ℝ which integrates miny2,1.For sake of clarity, we takeΘ is the open interval ℝ. As in the precited literature, we will assume the following.(i)
The existence of densitiesgtθ, such that θ↦gtθ is regular enough,(ii)
The convergence, asn⟶∞, of some integrals depending on gunθ.Theorem1 and Corollary 1 provide conditions ensuring the LAN property for the model (6), when the process X is observed along the discrete scheme (3). Denoting g¯unθ, the logarithmic derivative of gunθ relative to θ, the asymptotic Fisher information quantity at each θ should satisfy(7)Iθ=limn⟶∞∫g¯unθ2xgunθxdx∈0,∞.It is difficult to find Lévy processes fulfilling (7), and the reasons are numerous, for instance, the existence of the densities gtθ, the fact that they are not explicit in general, and their degeneracy as t⟶0. For these reasons, Corollary 1 focuses on the linear dependance (4) of the characteristics relative to θ. In this case, we may assume, without loss of generality, that Θ contains a reference value, 1 for example, and the value 0 is excluded in order to avoid trivialities. In this case, we only need to assume some regularities of the function gun1x and conditions of the kind (7) for θ=1. Let hn≔gun1 and h¯n be the logarithmic derivatives of hnx. The asymptotic Fisher information quantity should then satisfy(8)Iθ=1θ2limn⟶∞∫1+xh¯nx2hnxdx∈0,∞.The case of the discretization with constant pathun=u∈0,∞ is quite obvious since the scale model (5) becomes a regular i.i.d. one, that is, to say Iθ is finite and nonnull. If un⟶0, the situation is more intricate because hn degenerates when n⟶∞. It turns out that even the linear model (5) is falsely simple to handle. Intuitively, one looks at special Lévy processes Y attracted by stable processes on the sense of (10). The price to pay is to exhibit refined controls on the probability density function of Yt,t>0. In a second step, we restrict our attention to the scale model (5). For simplicity’s sake, it is easier in this case to express the probabilities ℙθθ∈Θ in the form (5) rather than considering them as solutions of martingale problems associated with the family of characteristics bθ,cθ,μθθ∈Θ because of the intricacy inherent in the truncation functions [13]. Generic examples of Lévy processes are stable processes. They characterized Lévy exponent as follows. Let α,β,γ,δ∈A=0,2×−1,1×0,+∞×ℝ and(9)Kα=α1l0,1α+α−21l1,2α,K¯α=Kαα.A stable process, with parametersα,β,γ,δ, is a Lévy process X=Stα,β,γ,δt≥0, such that the corresponding Lévy exponent is given by(10)φu=iδu−γuαexp−iπβKαSignu/2,ifα≠1,iδu−γu1+i2βloguSignu/π,ifα=1.The parameterα is the stability coefficient, β is the skewness coefficient, γ is the scale coefficient, and δ is the drift parameter. The corresponding triplet b,c,μ of characteristics is given by(11)b=δ+γsinπβKα/21lα≠1,c=2γ1lα=2,μdx=Cα,β,γ1lx>0+Cα,−β,γ1lx<0x−α+1dx1lα<2,Cα,β,γ=γπΓα+1sinπ2α1+K¯αβ,ifα≠1,1+β,ifα=1.See [14]. In model (6), a candidate for the unknown parameter θ could be any the parameters α,β,γ, or δ. Since stable processes enjoy the scaling property, with δα,t≔tδ+2β/πγlogγ1lα=1,(12)Stα,β,γ,δt≥0=dγ1/αδα,t+Stα,β,1,0t≥0,then a candidate for the unknown parameter in the model (5) is clearly the parameter θ=γ1/α. For more account on Lévy processes, the reader is referred to [13] or [15] and for stable distributions, we suggest [16] and also [17]. Section 4 provides nontrivial examples of LAN models associated with Lévy processes Y attracted by stable ones. That means that there exist measurable functions bt∈ℝ,at>0 and a nondegenerate distribution ν, such that(13)Y^t=Yt−btat⟶dν,ast⟶0or ast⟶∞.In [18], Far focused on the LAMN property for the model (5) discretized along the scheme iun=i/n,1≤i≤n, when the process Y is of the form Y=W+N, the sum of a standard Brownian motion and an independent compound Poisson process. She obtained LAMN property under the condition that the Lévy measure ν of N has no diffuse singular part and that if ν is absolutely continuous, then the model has the LAN property. Our development in Section 5 constitutes a complement to Corollary 1 for the scale model (5) and also to Far’s work [18] and illustrates how to build a LAN scale model from another LAN scale model.


## 2. Definition of the LAN Property

In Section 3, we provide some theoretical results on models associated with observations, at times iun,un>0, of the process X, and to illustrate by some examples. To this end, we consider the sequence of i.i.d. random variables and the family of σ-fields:(14)Xjn=Xj+1un−Xjun,Gin=σXjn,0≤j≤i−1=σjun,0≤j≤i.Denotingℋn=Gnn and ℋtn=Gntn,t∈0,1, we introduce the sequence of filtered statistical models:(15)En=Ω,ℋn,ℋtnt∈0,1,ℙθθ∈Θ.For any fixedθ0∈Θ, we denote(16)Θn=θ∈ℝ:θ0+θn∈Θ,θn=θ0+θn,ℙθn=ℙθn|ℋn,and we introduce the statistical experiments localized around θ0:(17)Enθ0=Ω,ℋn,ℋtnt∈0,1,ℙθnθ∈Θn,ℰ′θ0=Ω′,ℱ′,ℱt′t∈0,1,ℙθ′θ∈ℝ,where the last statistical experiment is a Gaussian Shift. By a Gaussian shift, we mean, that for all θ∈ℝ,ℙθ′ is the unique probability on Ω′,ℱ′ equivalent to ℙ0′ on each ℱt′ and that its associated likelihood process is the geometric Brownian motion defined by(18)Zt′θ=dℙθ′|ℱt′dℙθ0′|ℱt′=expθIθ0Xt′−θ22Iθ0t,t∈0,1,where Xt′t∈0,1 is a Wiener process, and then, under ℙθ′, the process Xt′−tθIθ0t≥0 is again a Wiener process. The quantity Iθ0 is called the asymptotic Fisher information quantity; it is a positive constant related to the sequence of statistical experiments Enθ0 in (17) and has to be determined. The asymptotic Fisher information quantity is crucial in any estimation procedure. Indeed, under the LAN property, 1/Iθ0 is the lower bound of the variance of any estimator ϑn of θ0. More precisely, HAJEK’s asymptotic convolution theorem says that if ϑn satisfies(19)ℒawnϑn−θ0+θn|ℙθn⟶ℒθ0,asn⟶∞,then the distribution ℒθ0 is the convolution product ℒθ0=ℒθ01∗ℒθ02, where ℒθ01=normal0,Iθ0−1 and ℒθ02 is a probability measure on ℝ. See [19] for more.Local asymptotic normality of the sequence of modelsEn in (17), in a value θ0∈Θ, is actually equivalent to the weak functional convergence in time of the sequence of statistical experiments Enθ0 to the Gaussian shift ℰ′θ0 in (17). This fact is explained as follows: let Z′ηξ and Zn,ηξ be the likelihood processes defined, for all η,ξ∈Θn and at each time t∈0,1, by(20)Zt′ηξ=dℙη′|ℱt′dℙξ′|ℱt′=Eℙξ′dℙη′dℙξ′|ℱt′=Zt′ηZt′ξ,Ztn,ηξ=dℙηn|ℋtndℙξn|ℋtn=Eℙξndℙηndℙξn|ℋtn,with the convention a/0=0, if a∈0,+∞. According to [5], the likelihood process Zn,ηξ of the statistical experiment Enθ0 is represented by(21)Ztn,ηξ=∏j=1ntgunηgunξXjn.The notion of weak functional convergence in time was introduced by Lecam [2] and developed by Strasser [19] and Jacod [20]. It is expressed as follows: for every finite subset J of ℝ=∪n≥1Θn, and every ξ∈Θ, we have(22)ℒawZn,ηξη∈J|ℙξnn⟶ℒawZ′ηξη∈J|ℙξ′,asn⟶+∞,in the sense of the weak convergence for the Skorohod topology.


## 3. When Does LAN Property Hold for Lévy Models?

Our aim is to give sufficient conditions on the p.d.f.>o of Xt under ℙθ, ensuring the LAN property for the sequence of filtered statistical models En.


### 3.1. LAN Property for the Model (6)

In this section, we will consider the model (6). If cθ=0, μθ integrates y∧1, and μθ0,∞=0 (respectively, μθ−∞,0=0), then the support of the distribution of Xt is(23)dθt,∞respectively,−∞,dθt,withdθ=bθ−∫ℝy1ly≤1μθdy.In all other cases, the distribution ofXt has a support equal to ℝ. There are many situations in which for all t>0,Xt has a probability p.d.f. gtθx which is infinitely differentiable in x. For instance, the latter holds if(24)cθ>0,or if∫y≤εminy2,1μθdy≥Kθεα,for any ε∈0,1 and for some Kθ>0 and some α∈0,2, see [21]. Later on, we may assume the following:(25)H0:for allθ∈Θandt>0,underℙθ,the support of the law ofXtis an intervalKt,independent fromθ,of the formKt=ℝor−∞,dtordt,∞,for somed∈ℝ,andXthas a p.d.f.x↦gtθxwhich is of classC2,relative toθ.We denoteKn=Kun and we define, on the interior of Kn, the following functions:(26)hnθ=gunθ,h¯nθ=∂∂θloghnθ,h¨nθ=∂2∂θ2hnθ,inθ=hnθh¯nθ2jnθ=inθ+h¨nθ.When the numberχ>0 appears, it is always understood that n is big enough so that χ and −χ are in Θn. For all θ∈Θ,ρ∈0,1, and ρ′=1−ρ, we denote(27)Inθ≔∫Kninθxdx,I˜nχ=supε≤χInεn,(28)J˜nρχ≔supζ,ε≤χ∫Knjnζnxjnεnxhnζnxρhnεnxρ′dx=J˜nρ′χ.For statisticians,Inθ0 is a familiar quantity and corresponds to the Fisher information quantity at stage n. The quantity J˜nρχ is less intuitive; it is a localized quantity around the true value θ0 and corresponds to the rest of Taylor approximations at the order 1 of Hellinger integrals of the model.We are now able to state our first result, that is, the LAN property for the model (6).Theorem 1.
Assume (H0) and the following conditions:(29)H1:limn⟶∞Inθ0=Iθ0∈0,∞and for allχ>0,limsupn⟶+∞I˜nχ<+∞,H2: There existsa∈0,1/2such that for allχ>0,one haslimn⟶+∞J˜nρχn=0forρ∈1/2,a,1−a.Then, the sequence of sequence of filtered statistical modelsEn (15), corresponding to (6), has the LAN property at θ0 with the speed n and the asymptotic Fisher information quantity Iθ0.Remark 1.
(i)
Cauchy–Schwarz inequality givesInθ≤J˜n1/2θ1/2, and both conditions (H1) and (H2) are implied by(30)H3:limn⟶+∞Inθ0=Iθ0and there existsa∈0,1/2, such that for allχ>0,one haslimsupn⟶+∞J˜nρχ<+∞forρ∈1/2,a,1−a.(ii)
Under different conditions and a different proof, Masuda obtained ([8], Theorem 2.12) the same conclusion as in Theorem 1.Genon-Catalot and Jacod [5] exhibited discretized models according to random sampling schemes Ti,n,1≤i≤n associated with a diffusion process X driven by Brownian motions (with coefficients dependent on θ and by an homogeneous way on X) and proved the LAMN property under conditions similar to (H0), that is, differentiability to the third order relative to θ and integrability of the densities of the processes. Their proofs have a general vocation in the sense that they only use the Markovian property of the processes and are based on a method of approximation of the log-likelihood. Because of the intricate form 16 of the likelihood processes, we show the weak functional convergence of Enθ0 to ℰ′θ0 via the convergence of the Hellinger processes, and with a tool, one can find in [20].Proof of Theorem1.
Fixθ0. The Hellinger process of order ρ∈0,1 between ℙη′ and ℙξ′, relative to ℱt′t∈0,1, is deterministic and has the form(31)ℋ′ηξρt=ρ1−ρη−ξ2Iθ0t2.
According to ([20], Theorem 5.3), it is enough to show that the Hellinger processes ℋn,ηξρ between ℙηn and ℙξn, relative to Gtnt∈0,1, satisfy the following: there exists a∈0,1/2, such that for every ∀η,ξ∈ℝ, ρ∈1/2,a,1−a, and t∈0,1, and the convergence in law(32)ℋn,ηξρt⟶ℙξnℋ′ηξρt,asn⟶∞,holds. We will use this method because in our framework, the processes ℋn,ηξ are also deterministic and have the following quite simple form one can find in [5]: with ⌊nt⌋ being the integer part of nt, we have(33)ℋn,ηξρt=nt1−∫Knhnηnρhnξn1−ρydy.
(1) Forρ∈0,1, ρ′=1−ρ, take Φρu,v=ρu+ρ′v−uρvρ′,u,v≥0, and observe that(34)ℋn,ηξρ1=∫KnΦρhnηn,hnξnydy.
According to (32) and (33), it is enough to show that ∀η,ξ∈ℝ, and ρ∈1/2,a,1−a, and we have(35)limn⟶∞n∫KnΦρhnηn,hnξnydy=ρρ′2η−ξ2Iθ0.
(2) Assume (H0), (H1), and (H2) for a fixeda∈0,1/2. Applying Taylor expansion at the first order of θ↦hnθρ for η∈ℝ and n big enough, we get for η∈Θn, the representation of hnηnρ on Kn:(36)hnηnρ=hnθ0ρ+ρηnknθ0,ρ+ρηnVnη,ρ,where for all θ∈Θ, the functions(37)knθ,ρ=h¯nθhnθρ,k˙nθ,ρ=hnθρ∂∂θh¯nθ+ρh¯nθ2=hnθρh¨nθhnθ−ρ′h¯nθ2,(38)Vnη,ρ=∫01knηrn,ρ−knθ0,ρdr=ηn∫011−rk˙nηrn,ρdr,are defined on the interior of Kn. Also, observe the following relations:(39)knθ,ρhnθρ′=knθ,ρ′hnθρ=knθ,1,knθ,ρknθ,ρ′=inθ,k˙nθ,ρhnθρ′=k˙nθ,1−ρ′inθ.
Because of (38) and (39), one has(40)Vnη,1−hnθ0ρ′Vnη,ρ=ηn∫011−rk˙nηrn,1−hnθ0ρ′k˙nηrn,ρdr=ηn∫011−rk˙nηrn,ρhnηrnρ′−hnθ0ρ′dr+ηρ′n∫011−rinηrndr.
Using (36), finally write(41)Vnη,1−hnθ0ρ′Vnη,ρ=η2ρ′n∫011−rk˙nηrn,ρknθ0,ρ′+Vnηr,ρ′dr+ηρ′n∫011−rinηrndr.
(3) For allη,ξ∈ℝ and n large enough, so that η,ξ∈Θn, write(42)Φρhnηn,hnξn=ρhnθ0+ηnknθ0,1+ηnVnη,1+ρ′hnθ0+ξnknθ0,1+ξnVnξ,1−hnθ0ρ+ρηnknθ0,ρ+ρηnVnη,ρhnθ0ρ′+ρ′ξnknθ0,ρ′+ρ′ξnVnξ,ρ′.
Then, use (39) and obtain(43)nΦρhnηn,hnξn=nρηVnη,1−hnθ0ρ′Vnη,ρ+nρ′ξVnξ,1−hnθ0ρVnη,ρ′−ρρ′ηξinθ0+Vnη,ρVnξ,ρ′+knθ0,ρVnξ,ρ′+knθ0,ρ′Vnη,ρ.
According to (41), we have(44)nΦρhnηn,hnξn=ρρ′Anη,ξ+Bnη,ξ,where(45)Anη,ξ=η2∫011−rinηrndr+ξ2∫011−rinξrndr−ηξinθ0,Bnη,ξ=η3n∫011−rk˙nηrn,ρknθ0,ρ′+Vnηr,ρ′dr+ξ3n∫011−rk˙nξrn,ρ′knθ0,ρ+Vnξr,ρdr,−ηξknθ0,ρVnξ,ρ′+knθ0,ρ′Vnη,ρ+Vnη,ρVnξ,ρ′.
(4a) We will now prove the following convergence that will imply (35): for all η,ξ∈ℝ,(46)limn⟶∞∫KnAnη,ξxdx=12η−ξ2Iθ0,(47)limn⟶∞∫KnBnη,ξxdx=0.
(4b). To prove (46), we use both (26) and (37), and for all ρ∈0,1 and θ∈Θ, we have the representation and the control.(48)knθ,ρ=inθhnθρ−1/2,k˙nθ,ρ≤jnθhnθρ′.
Letη,ξ∈ℝ, and let χ=η∨ξ. Using (26), (37), and Taylor expansion at the first order of θ↦knθ,1/22, we obtain the galloping control, valid for all r∈0,1:(49)Inηrn−Inθ0=∫Knknηrn,1/2x2−knθ0,1/2x2dx=2ηrn∫∫01knηrsn,1/2xk˙nηrsn,1/2xdsdx.
Takingρ=1/2 in (48), applying Cauchy–Schwarz inequality and assuming (H1) and (H2), we obtain(50)supr∈0,1Inηrn−Inθ0≤2ηI˜nχJ˜n1/2χn1/2⟶0asn⟶∞.
We then can write(51)∫KnAnη,ξxdx=η2∫011−rInηrndr+ξ2∫011−rInξrndr−ηξInθ0=η2∫011−rInηrn−Inθ0dr+ξ2∫011−rInξrn−Inθ0dr+12η2+ξ2−2ηξInθ0,and clearly (50) implies (46).
(4c). To prove (47), we use same the same arguments as in (4b), the Taylor expansion at the first order of θ↦knθ,1/2, and the representation(52)k˙nηrn,ρknθ0,ρ′=k˙nηrn,ρknηrn,ρ′−ηrn∫01k˙nηrn,ρk˙nηrsn,ρ′ds,ρ∈0,1,r∈0,1.
By (48) and Cauchy–Schwarz inequality, we have the following control, valid for all r∈0,1:(53)∫Knk˙nηrn,ρxknθ0,ρ′xdx≤∫Kninηrnxjnηrnxhnηrnxdx+ηrnJ˜nρχ.
The latter implies(54)supr∈0,1∫Knk˙nηrn,ρxknθ0,ρ′xdx≤I˜nχJ˜n1/2χ+χnJ˜nρχ≔δnρχ=δnρ′χ.
Using (48), we also have(55)∫Knk˙nηrn,ρxVnηr,ρ′xdx≤ηrn∫011−s∫Knk˙nηrn,ρxk˙nηrsn,ρ′xdxds,and reproducing the method, we used to get (54), and we obtain(56)supr∈0,1∫Knk˙nηrn,ρxVnηr,ρ′xdx≤δnρχ.
According to (38) and (54), we also have(57)∫Knknθ0,ρxVnη,ρ′xdx≤ηn∫011−r∫Knk˙nηrn,ρ′xknθ0,ρxdxdr≤χnδnρ′χ.
Furthermore, (38), (48), and Cauchy–Schwarz inequality imply(58)∫KnVnη,ρxVnξ,ρ′xdx≤ηξn∫01∫011−r1−s∫jnηrnxjnξsnxhnηrnxρ′hnξsnxρdxdsdr≤χnδnρ′χ.
Finally, according to (45), (54), (56)–(58), we obtain the control(59)∫KnBnη,ξxdx≤7χ3δnρχn=7χ3I˜nχJ˜n1/2χn1/2+χJ˜nρχn,and we conclude with the fact that assumptions (H1) and (H2) imply (47).


### 3.2. LAN Property for the Scale Model (5)

As a consequence of Theorem1, we obtain a result for the scale model (5). The main argument is that the r.v. Xt,t>0 has a p.d.f gt1 under ℙ1 if and only if Xt has the p.d.f. gtθx=gt1x/θ/θ under ℙθ. From now on, the functions hn,hn′, and hn″ denote, respectively, hn1, the p.d.f. of Xun under ℙ1, the first and the second derivatives of hn. For θ∈Θ and n≥1, we have the expression(60)hnθx=1θhnxθ,xθ∈Kn≔supporthn.Therefore, if we want (H0) to be satisfied, we need to imposeK=Kn=ℝorℝ+orℝ−. Then, for all x∈K, we have(61)inθx=1θ3inxθ,jnθx=1θ3jnxθ,where the functions in≔in1 and jn≔jn1 are given by(62)inx=1+xhn′hnx2hnx,jnx=inx+2+4xhn′hnx+x2hn″hnxhnx.Making a change of variables, we see that the quantitiesInθ and I˜nχ, defined in (27), satisfy(63)Inθ=1θ2∫Kinxdx=Inθ2,I˜nχ=supζ≤χInζn2≤Inθ0−χ/n2.Using Cauchy–Schwarz inequality and again a change of variables, we obtain the following: for allχ>0, the quantities J˜nρχ,ρ∈0,1, defined in (28), satisfy(64)J˜nρχ≤supζ,ξ≤χ∫Kjnζnx2hnζnx2ρdx1/2∫Kjnεnx2hnεnx2ρ′dx1/2=supζ,ε≤χ1ζn5−2ρεn5−2ρ∫Kjnx2hnx2ρdx∫Kjnx2hnx2ρ′dx1/2≤θ0−χn4ρ−10JnρJnρ′1/2,Jnr≔∫Kjnx2hnx2rdx.Corollary 1.
For the scale model (5), the sequence of statistical models En has the LAN property with speed n at each θ∈Θ if the following conditions are satisfied:(65)C0: underℙ1,the supportKof the law ofX1is eitherℝorℝ+orℝ−.For allt>0,the r.v.Xthas a p.d.f.gt1xwhich is of classC2on the interior ofK.C1:the sequenceIn≔∫Kinxdxsatisfieslimn⟶+∞In=I∈0,∞,C2:there existsa∈0,12such thatlimn⟶∞1nJnρ=0forρ∈12,a,1−a.
In this case, the asymptotic Fisher information quantity isIθ=I/θ2.Proof.
It is immediate that (C0) implies (H0). Representation (63) shows that (C1) implies (H1) and (64) shows that (C2) implies (H2).Remark 2.
Sincehn is the p.d.f. of Xun, then we should be aware that the discretization path un intervene in the assumptions (C1) and (C2). In Theorem 2, we will see that in some favorable cases, the effect of the path un has a quite negligible effect. In general, the LAN property depends strongly on the asymptotic of un.


## 3.1. LAN Property for the Model (6)

In this section, we will consider the model (6). If cθ=0, μθ integrates y∧1, and μθ0,∞=0 (respectively, μθ−∞,0=0), then the support of the distribution of Xt is(23)dθt,∞respectively,−∞,dθt,withdθ=bθ−∫ℝy1ly≤1μθdy.In all other cases, the distribution ofXt has a support equal to ℝ. There are many situations in which for all t>0,Xt has a probability p.d.f. gtθx which is infinitely differentiable in x. For instance, the latter holds if(24)cθ>0,or if∫y≤εminy2,1μθdy≥Kθεα,for any ε∈0,1 and for some Kθ>0 and some α∈0,2, see [21]. Later on, we may assume the following:(25)H0:for allθ∈Θandt>0,underℙθ,the support of the law ofXtis an intervalKt,independent fromθ,of the formKt=ℝor−∞,dtordt,∞,for somed∈ℝ,andXthas a p.d.f.x↦gtθxwhich is of classC2,relative toθ.We denoteKn=Kun and we define, on the interior of Kn, the following functions:(26)hnθ=gunθ,h¯nθ=∂∂θloghnθ,h¨nθ=∂2∂θ2hnθ,inθ=hnθh¯nθ2jnθ=inθ+h¨nθ.When the numberχ>0 appears, it is always understood that n is big enough so that χ and −χ are in Θn. For all θ∈Θ,ρ∈0,1, and ρ′=1−ρ, we denote(27)Inθ≔∫Kninθxdx,I˜nχ=supε≤χInεn,(28)J˜nρχ≔supζ,ε≤χ∫Knjnζnxjnεnxhnζnxρhnεnxρ′dx=J˜nρ′χ.For statisticians,Inθ0 is a familiar quantity and corresponds to the Fisher information quantity at stage n. The quantity J˜nρχ is less intuitive; it is a localized quantity around the true value θ0 and corresponds to the rest of Taylor approximations at the order 1 of Hellinger integrals of the model.We are now able to state our first result, that is, the LAN property for the model (6).Theorem 1.
Assume (H0) and the following conditions:(29)H1:limn⟶∞Inθ0=Iθ0∈0,∞and for allχ>0,limsupn⟶+∞I˜nχ<+∞,H2: There existsa∈0,1/2such that for allχ>0,one haslimn⟶+∞J˜nρχn=0forρ∈1/2,a,1−a.Then, the sequence of sequence of filtered statistical modelsEn (15), corresponding to (6), has the LAN property at θ0 with the speed n and the asymptotic Fisher information quantity Iθ0.Remark 1.
(i)
Cauchy–Schwarz inequality givesInθ≤J˜n1/2θ1/2, and both conditions (H1) and (H2) are implied by(30)H3:limn⟶+∞Inθ0=Iθ0and there existsa∈0,1/2, such that for allχ>0,one haslimsupn⟶+∞J˜nρχ<+∞forρ∈1/2,a,1−a.(ii)
Under different conditions and a different proof, Masuda obtained ([8], Theorem 2.12) the same conclusion as in Theorem 1.Genon-Catalot and Jacod [5] exhibited discretized models according to random sampling schemes Ti,n,1≤i≤n associated with a diffusion process X driven by Brownian motions (with coefficients dependent on θ and by an homogeneous way on X) and proved the LAMN property under conditions similar to (H0), that is, differentiability to the third order relative to θ and integrability of the densities of the processes. Their proofs have a general vocation in the sense that they only use the Markovian property of the processes and are based on a method of approximation of the log-likelihood. Because of the intricate form 16 of the likelihood processes, we show the weak functional convergence of Enθ0 to ℰ′θ0 via the convergence of the Hellinger processes, and with a tool, one can find in [20].Proof of Theorem1.
Fixθ0. The Hellinger process of order ρ∈0,1 between ℙη′ and ℙξ′, relative to ℱt′t∈0,1, is deterministic and has the form(31)ℋ′ηξρt=ρ1−ρη−ξ2Iθ0t2.
According to ([20], Theorem 5.3), it is enough to show that the Hellinger processes ℋn,ηξρ between ℙηn and ℙξn, relative to Gtnt∈0,1, satisfy the following: there exists a∈0,1/2, such that for every ∀η,ξ∈ℝ, ρ∈1/2,a,1−a, and t∈0,1, and the convergence in law(32)ℋn,ηξρt⟶ℙξnℋ′ηξρt,asn⟶∞,holds. We will use this method because in our framework, the processes ℋn,ηξ are also deterministic and have the following quite simple form one can find in [5]: with ⌊nt⌋ being the integer part of nt, we have(33)ℋn,ηξρt=nt1−∫Knhnηnρhnξn1−ρydy.
(1) Forρ∈0,1, ρ′=1−ρ, take Φρu,v=ρu+ρ′v−uρvρ′,u,v≥0, and observe that(34)ℋn,ηξρ1=∫KnΦρhnηn,hnξnydy.
According to (32) and (33), it is enough to show that ∀η,ξ∈ℝ, and ρ∈1/2,a,1−a, and we have(35)limn⟶∞n∫KnΦρhnηn,hnξnydy=ρρ′2η−ξ2Iθ0.
(2) Assume (H0), (H1), and (H2) for a fixeda∈0,1/2. Applying Taylor expansion at the first order of θ↦hnθρ for η∈ℝ and n big enough, we get for η∈Θn, the representation of hnηnρ on Kn:(36)hnηnρ=hnθ0ρ+ρηnknθ0,ρ+ρηnVnη,ρ,where for all θ∈Θ, the functions(37)knθ,ρ=h¯nθhnθρ,k˙nθ,ρ=hnθρ∂∂θh¯nθ+ρh¯nθ2=hnθρh¨nθhnθ−ρ′h¯nθ2,(38)Vnη,ρ=∫01knηrn,ρ−knθ0,ρdr=ηn∫011−rk˙nηrn,ρdr,are defined on the interior of Kn. Also, observe the following relations:(39)knθ,ρhnθρ′=knθ,ρ′hnθρ=knθ,1,knθ,ρknθ,ρ′=inθ,k˙nθ,ρhnθρ′=k˙nθ,1−ρ′inθ.
Because of (38) and (39), one has(40)Vnη,1−hnθ0ρ′Vnη,ρ=ηn∫011−rk˙nηrn,1−hnθ0ρ′k˙nηrn,ρdr=ηn∫011−rk˙nηrn,ρhnηrnρ′−hnθ0ρ′dr+ηρ′n∫011−rinηrndr.
Using (36), finally write(41)Vnη,1−hnθ0ρ′Vnη,ρ=η2ρ′n∫011−rk˙nηrn,ρknθ0,ρ′+Vnηr,ρ′dr+ηρ′n∫011−rinηrndr.
(3) For allη,ξ∈ℝ and n large enough, so that η,ξ∈Θn, write(42)Φρhnηn,hnξn=ρhnθ0+ηnknθ0,1+ηnVnη,1+ρ′hnθ0+ξnknθ0,1+ξnVnξ,1−hnθ0ρ+ρηnknθ0,ρ+ρηnVnη,ρhnθ0ρ′+ρ′ξnknθ0,ρ′+ρ′ξnVnξ,ρ′.
Then, use (39) and obtain(43)nΦρhnηn,hnξn=nρηVnη,1−hnθ0ρ′Vnη,ρ+nρ′ξVnξ,1−hnθ0ρVnη,ρ′−ρρ′ηξinθ0+Vnη,ρVnξ,ρ′+knθ0,ρVnξ,ρ′+knθ0,ρ′Vnη,ρ.
According to (41), we have(44)nΦρhnηn,hnξn=ρρ′Anη,ξ+Bnη,ξ,where(45)Anη,ξ=η2∫011−rinηrndr+ξ2∫011−rinξrndr−ηξinθ0,Bnη,ξ=η3n∫011−rk˙nηrn,ρknθ0,ρ′+Vnηr,ρ′dr+ξ3n∫011−rk˙nξrn,ρ′knθ0,ρ+Vnξr,ρdr,−ηξknθ0,ρVnξ,ρ′+knθ0,ρ′Vnη,ρ+Vnη,ρVnξ,ρ′.
(4a) We will now prove the following convergence that will imply (35): for all η,ξ∈ℝ,(46)limn⟶∞∫KnAnη,ξxdx=12η−ξ2Iθ0,(47)limn⟶∞∫KnBnη,ξxdx=0.
(4b). To prove (46), we use both (26) and (37), and for all ρ∈0,1 and θ∈Θ, we have the representation and the control.(48)knθ,ρ=inθhnθρ−1/2,k˙nθ,ρ≤jnθhnθρ′.
Letη,ξ∈ℝ, and let χ=η∨ξ. Using (26), (37), and Taylor expansion at the first order of θ↦knθ,1/22, we obtain the galloping control, valid for all r∈0,1:(49)Inηrn−Inθ0=∫Knknηrn,1/2x2−knθ0,1/2x2dx=2ηrn∫∫01knηrsn,1/2xk˙nηrsn,1/2xdsdx.
Takingρ=1/2 in (48), applying Cauchy–Schwarz inequality and assuming (H1) and (H2), we obtain(50)supr∈0,1Inηrn−Inθ0≤2ηI˜nχJ˜n1/2χn1/2⟶0asn⟶∞.
We then can write(51)∫KnAnη,ξxdx=η2∫011−rInηrndr+ξ2∫011−rInξrndr−ηξInθ0=η2∫011−rInηrn−Inθ0dr+ξ2∫011−rInξrn−Inθ0dr+12η2+ξ2−2ηξInθ0,and clearly (50) implies (46).
(4c). To prove (47), we use same the same arguments as in (4b), the Taylor expansion at the first order of θ↦knθ,1/2, and the representation(52)k˙nηrn,ρknθ0,ρ′=k˙nηrn,ρknηrn,ρ′−ηrn∫01k˙nηrn,ρk˙nηrsn,ρ′ds,ρ∈0,1,r∈0,1.
By (48) and Cauchy–Schwarz inequality, we have the following control, valid for all r∈0,1:(53)∫Knk˙nηrn,ρxknθ0,ρ′xdx≤∫Kninηrnxjnηrnxhnηrnxdx+ηrnJ˜nρχ.
The latter implies(54)supr∈0,1∫Knk˙nηrn,ρxknθ0,ρ′xdx≤I˜nχJ˜n1/2χ+χnJ˜nρχ≔δnρχ=δnρ′χ.
Using (48), we also have(55)∫Knk˙nηrn,ρxVnηr,ρ′xdx≤ηrn∫011−s∫Knk˙nηrn,ρxk˙nηrsn,ρ′xdxds,and reproducing the method, we used to get (54), and we obtain(56)supr∈0,1∫Knk˙nηrn,ρxVnηr,ρ′xdx≤δnρχ.
According to (38) and (54), we also have(57)∫Knknθ0,ρxVnη,ρ′xdx≤ηn∫011−r∫Knk˙nηrn,ρ′xknθ0,ρxdxdr≤χnδnρ′χ.
Furthermore, (38), (48), and Cauchy–Schwarz inequality imply(58)∫KnVnη,ρxVnξ,ρ′xdx≤ηξn∫01∫011−r1−s∫jnηrnxjnξsnxhnηrnxρ′hnξsnxρdxdsdr≤χnδnρ′χ.
Finally, according to (45), (54), (56)–(58), we obtain the control(59)∫KnBnη,ξxdx≤7χ3δnρχn=7χ3I˜nχJ˜n1/2χn1/2+χJ˜nρχn,and we conclude with the fact that assumptions (H1) and (H2) imply (47).


## 3.2. LAN Property for the Scale Model (5)

As a consequence of Theorem1, we obtain a result for the scale model (5). The main argument is that the r.v. Xt,t>0 has a p.d.f gt1 under ℙ1 if and only if Xt has the p.d.f. gtθx=gt1x/θ/θ under ℙθ. From now on, the functions hn,hn′, and hn″ denote, respectively, hn1, the p.d.f. of Xun under ℙ1, the first and the second derivatives of hn. For θ∈Θ and n≥1, we have the expression(60)hnθx=1θhnxθ,xθ∈Kn≔supporthn.Therefore, if we want (H0) to be satisfied, we need to imposeK=Kn=ℝorℝ+orℝ−. Then, for all x∈K, we have(61)inθx=1θ3inxθ,jnθx=1θ3jnxθ,where the functions in≔in1 and jn≔jn1 are given by(62)inx=1+xhn′hnx2hnx,jnx=inx+2+4xhn′hnx+x2hn″hnxhnx.Making a change of variables, we see that the quantitiesInθ and I˜nχ, defined in (27), satisfy(63)Inθ=1θ2∫Kinxdx=Inθ2,I˜nχ=supζ≤χInζn2≤Inθ0−χ/n2.Using Cauchy–Schwarz inequality and again a change of variables, we obtain the following: for allχ>0, the quantities J˜nρχ,ρ∈0,1, defined in (28), satisfy(64)J˜nρχ≤supζ,ξ≤χ∫Kjnζnx2hnζnx2ρdx1/2∫Kjnεnx2hnεnx2ρ′dx1/2=supζ,ε≤χ1ζn5−2ρεn5−2ρ∫Kjnx2hnx2ρdx∫Kjnx2hnx2ρ′dx1/2≤θ0−χn4ρ−10JnρJnρ′1/2,Jnr≔∫Kjnx2hnx2rdx.Corollary 1.
For the scale model (5), the sequence of statistical models En has the LAN property with speed n at each θ∈Θ if the following conditions are satisfied:(65)C0: underℙ1,the supportKof the law ofX1is eitherℝorℝ+orℝ−.For allt>0,the r.v.Xthas a p.d.f.gt1xwhich is of classC2on the interior ofK.C1:the sequenceIn≔∫Kinxdxsatisfieslimn⟶+∞In=I∈0,∞,C2:there existsa∈0,12such thatlimn⟶∞1nJnρ=0forρ∈12,a,1−a.
In this case, the asymptotic Fisher information quantity isIθ=I/θ2.Proof.
It is immediate that (C0) implies (H0). Representation (63) shows that (C1) implies (H1) and (64) shows that (C2) implies (H2).Remark 2.
Sincehn is the p.d.f. of Xun, then we should be aware that the discretization path un intervene in the assumptions (C1) and (C2). In Theorem 2, we will see that in some favorable cases, the effect of the path un has a quite negligible effect. In general, the LAN property depends strongly on the asymptotic of un.


## 4. Examples of LAN Property for Lévy Processes Attracted by Stable Processes

In this section, we provide some examples of Lévy processesX satisfying the conditions of Corollary 1. This corollary provides the LAN property for the scale model (5), X=θY, under regularity and integrability conditions on the p.d.f. of Yt,t>0. Unfortunately, these p.d.f., if they exist, are not explicit in general; for this reason, we focus on processes Y which belong to the domain of attraction of stable processes.We recall that a processSα,β,γ,δ is stable, if it is a Lévy process with Lévy exponent given in (10). In the case α∈0,1 and δ≥0, the process Sα,1,γ,δ is a subordinator, i.e., a positive increasing the Lévy process, and the distribution of Stα,1,γ,δ,t>0, has a support equal to δt,∞. In all cases, Stα,1,γ,δ,t>0, has a p.d.f. Gtα,β,γ,δ which is infinitely differentiable and is explicit only for the particular values α,β=1/2,1,1,0,2,0. These values correspond, respectively, to the first passage times of the Brownian motion, the Cauchy process, and the Brownian Motion. Otherwise, Gtα,β,γ,δ is be expressed only as the inverse Fourier transform of expϕu. As announced, the introduction, we focus in this section on the case where the Lévy processes Y, in the scale model (5), is attracted by a stable process, i.e., satisfying (13). Bertoin and Doney [22] showed that if (10) holds, then the process Y is attracted by a stable one, and then, the following holds:(i)
There existsb∈ℝ, α∈0,2, and a slowly varying function lt (i.e., a function satisfying lλt/lt⟶1,∀λ>0), such that bt=bt and at=t1/αlt(ii)
νdx=Gxdx, where G is the p.d.f. of some stable r.v. S1α,β,γ,0,β∈−1,1,γ>0;(iii)
The p.d.f.Gt of Y^t,t>0, is infinitely differentiable and satisfies(66)Gtkx⟶Gkx,uniformly inx∈SupportG,t⟶0ort⟶∞,k∈ℕ.(iv)
The convergence in (13) can be entirely expressed with the behavior of the tail of the Lévy measure of Y or by the existence of a Brownian component.Observe that the p.d.f.hn of Yun is represented by(67)hx=Gunx−bunaun,x∈ℝ.As in (8), let us denote by G¯un and G¯ the logarithmic derivative of Gun and logG, respectively. After the change of variable x↦aunx+bun, the asymptotic Fisher information quantity in (63) takes the form(68)Iθ=Iθ2,I=limn⟶∞∫1+x+bunaunG¯unx2Gunxdx.The convergence (66) does not ensure that Iθ is finite and nonnull; it only ensures that if it happens, then(69)l=limn⟶∞bunaunexists,I=∫1+x+lG¯x2Gxdx∈0,∞.The stronger control(70)limsupx⟶∞,n⟶∞xG¯unx<∞,is sufficient to prove (69), but it is hard to obtain. In [14, 17], we provided controls for the p.d.f. Gtx in the space variable x and uniformly in small or big time t, the processes(71)Y=∑i=kNSak,bk,ck,dk,orY=Sα,β,1,0∘Z,where the coefficients ak,bk,ck,dk vary in a set DN, the processes Sak,bk,ck,dk are independent, and Sα,β,1 is independent of the subordinator Z. We emphasize that in (71), Y satisfies (13) and does not enjoy the scaling property (12) anymore.(i)
IfY=∑i=kNSak,bk,ck,dk and the stability coefficients ak are pairwise different, we have an asymptotic scaling property; if i∧=Argminαi,1≤i≤N and i∨=Argmaxαi,1≤i≤N and i=i∨ (respectively, i∧), then(72)Yt−∑k=1Nδt,akγit1/ai⟶dS1ai,bi,1,0,ast⟶0respectively,t⟶∞.The last property will be useful in Subsection4.2 when taking the path un⟶0 (respectively, un⟶∞).(ii)
IfY=Sα,β,1,0∘Z, we assume that the subordinator Z is itself attracted by a stable subordinator: for some 0<ϵ<1 and some (deterministic) speed rt,(73)Ztrt⟶dS1ε,1,1,0,ast⟶0respectively, as∞.The functiont↦rt is necessarily a regularly varying function of order 1/ϵ at 0 (respectively, ∞): rt=t1/ϵlt and l is slowly varying, i.e., lλt/lt⟶1,∀λ>0. Then, there exist β∈−1,1 and γ′>0, such that(74)Ytrt1/α⟶dS1εα,β′,γ′,0,ast⟶0respectively,t⟶∞.


### 4.1. The Scale Models Associated with Stable Processes Have the LAN Property

The following result is a first application of Corollary1.Theorem 2.
For the scale model (5), assume Y=Sα,β,γ,δ is a stable process, such that δ=0 is null if α<1 and β=1. Recall δt,α is defined in (12). Let the path un satisfy ln=γun−1/αδun,α⟶l∈ℝ and one of the following conditions:(i)
un⟶L∈0,∞(ii)
un⟶L=0 and nRun⟶∞, for some R>0(iii)
un⟶L=+∞ and n−Sun⟶0, for some S>0
Then, the sequence of sequence of filtered statistical scale modelsEn (15) have the LAN property with speed n at each θ∈Θ. The asymptotic Fisher information quantity is given by(75)Iθ=1θ2∫y+l2G′y2Gydy−1,where G is the p.d.f. of S1α,β,1,0.Remark 3.
Ifα∈0,1, then support of the law of S1α,β,γ,δ is δ,∞ (respectively, −∞,δ) if β=1 (respectively, β=−1); otherwise, the support is whole ℝ. The assumption δ=0 ensures that the support of the law of S1α,β,γ,δ satisfies the condition (C0) of Corollary 1. We are aware that the cases nun⟶0andun⟶∞ are not very realistic by the statistical point of view. The most interesting cases are as follows:(a)
un=u∈0,∞, then L=u, and the assumptions of Theorem 2 are satisfied. This is an essentially trivial result because we treat then a regular i.i.d. model.(b)
un=u/n,u∈0,∞, then L=0, and the assumptions of Theorem 2 are satisfied if and only if one of the following holds:(i)
α<1,δ=0, and then, l=0;(ii)
α=1,β=0, and then, l=δ/γ;(iii)
α>1, and then, l=0.Proof.
of Theorem2. (1a) Recall that hn denotes the infinitely differentiable p.d.f of Sunα,β,γ,δ. Due to the scaling property (12), we have the following representations: for all k∈ℕ and x∈supporthn,(76)hnkx=1γunk+1/αG1α,β,1,0kx−δun,αγun1/α,where(77)Supporthn=ℝ+,if0<α<1,β=1,ℝ−,if0<α<1,β=−1,ℝ,otherwise.
(1b) In [14, 17], we provided several properties of the p.d.f. G=G1α,β,1,0. For instance, there exist positive constants A,B,C,D that depend explicitly on α,β, such that(78)Gx∼0+respectively,∼0−ξx,ifβ=1respectively,−1and0<α<1,∼+∞respectively,∼−∞χx,ifβ≠−1respectively,1and0<α<2,∼+∞respectively,∼−∞ηx,ifβ=−1respectively,1andα=1,∼+∞respectively,∼−∞ξx,ifβ=−1respectively,1and1<α≤2.where(79)χx≔Dxα+1,ξx≔Bx2−α/2α−1e−Axα/α−1,ηx=Cexp−eπx/2+πx4,and fx∼lgx means limx⟶lfx/gx=1. Furthermore, with the convention 0/0=0, the functions(80)Fkx≔GkxGx,k∈ℕ,are continuous on supporthn, given by (77), and there exist positive numbers a,b,c, depending explicitly on α and β and k, such that(81)Fkx∼0+respectively,∼0−axk/α−1,ifβ=1respectively,−1and0<α<1,∼+∞respectively,∼−∞bx−k,ifβ≠−1respectively,1and0<α<2,∼+∞respectively,∼−∞cexpkπx2,ifβ=−1respectively,1andα=1,∼+∞respectively,∼−∞axk/α−1,ifβ=−1respectively,1and1<α≤2.
From the last equivalences, we see that for any nonnegative integers, we have the implication(82)0≤r≤k⟹limx⟶∞x21−ρ1+αxrFksG21−ρ∈0,∞.
Thus, since0<ρ<1−1/21+α⇔21−ρ1+α>1, then(83)0≤r≤k,0<ρ<1−1α+1⟹x↦xrFkxsGx21−ρ∈L1dx.
(2) We need to verify the assumptions of Corollary1, i.e., to check (C1):(84)In=∫1+xhn′hnx2hnxdx⟶I=∫y+l2G′2Gydy−1∈0,∞,asn⟶∞,and also (C2); there exists a∈0,1/2, such that for ρ∈1/2,a,1−a, one has(85)1nJnρ=1n∫jn2hn2ρxdx⟶0,jnx=1+xhn′hnx2+2+4xhn′hnx+x2hn″hnxhnx.
(3) The scaling property (76) and the corresponding change of variables give the following representation of the quantity In in (84):(86)In=∫1+y+lnG′yGy2Gydy.
Using (82) and the fact that ln⟶l∈ℝ, we obtain(87)x↦supn∈ℕ1+y+lnG′Gy2Gy∈L1dy,limn⟶∞In=∫1+y+lG′Gy2Gydy.
Developing the last expression, integrating by parts and using the fact thatGy and yGy both tend to 0 as y goes to each endpoint of the support (77), we recover (84).
(4) Again, by the change of variables corresponding to (76) and by the representation (85), one has(88)Jnρ=γun2ρ−1/α∫1+y+lnG′Gy2+2+4y+lnG′Gy+y+ln2G″Gy2Gy21−ρdy.
Thanks to (83) and to the fact that ln⟶l∈ℝ, and it is clear that if ρ∈0,1−1/2α+1, then(89)supn∈ℕ1+y+lnG′Gy2+2+4y+lnG′Gy+y+ln2G″Gy2Gy21−ρis integrable. Let ϵ≔1/2−1/2α+1. To prove the convergence (85), it is enough to have(90)ρ∈0,ϵ+12,(91)limx⟶∞un2ρ−1/αn=0.
(5) Now, we distinguish between the values ofL=limn⟶∞un.(a)
IfL=0 and if ρ≥1/2, then (91) is true. By assumptions, we have 1/un≤nR, for large n. If ρ<1/2, then we have (91) as soon as R1−2ρ/α<1, which is equivalent to ρ>1/2−α/2R. Thus, we only have to choose ϵ′=ϵ∧α/4R and a=1/2−ϵ′ to get (90) to obtain (91) for ρ∈1/2,a,1−a.(b)
IfL∈0,∞, then (91) is always true.(c)
IfL=+∞ and if ρ≤1/2, then (91) is true. As in (5a), we have un≤nS for large n. If ρ>1/2 then we have (91) as soon as S2ρ−1/α<1, which is equivalent to ρ<1/2+α/2S. Thus, we only need to choose ϵ″=ϵ∧α/4Sanda=1/2−ϵ″ to obtain (90) and (91) for ρ∈1/2,a,1−a.


### 4.2. The Scale Models Associated with the Sum of Independent Stables Processes Have the LAN Property

This subsection gives a second example which also generalizes the previous one and achieves the situation (13). Define K¯α=1 if α≤1 and K¯α=α−2/α and assume the following restrictions on the skewness parameters:(92)Sa,b: we haveNindependent stable processesSak,bk,ck,0,such that the parameters satisfyaa1<a2<⋯<aN<2andD=∩k=1Nak,21+bkK¯ak≠∅;bbk=0ifak=1andB=maxbkK¯akK¯aN,1≤k≤N<1.LetY=∑k=1NSak,bk,ck,0 and Yi be the processes defined by(93)Yti=Ytt1/ai,i=1ori=Nand,t>0.The processesYi satisfy the asymptotic scaling property (72). Denote by Ht the p.d.f of Yt,(94)Ht=Gta1,b1,c1,0∗⋯∗GtaN,bN,cN,0,and by Hi,t, the one of Yti. Then, hn=Hun satisfies(95)hnkx=unci−k+1/αiHi,unkunci−1/aix.In [14, 17], we showed that the following functions H0 and H∞ given by(96)HL≔limt⟶∞H1,tx=G1a1,b1,1,0x,ifL=+∞,limt⟶0+HN,tx=G1aN,bN,1,0x,ifL=0,are well defined, and the above convergence holds uniformly in x∈supportG1ai,bi,1,0 and still hold for the successive derivatives. As one can guess, we are going to exploit identity (95) and state the following result.Theorem 3.
Letunn be a sequence satisfying one of the following conditions:(i)
un⟶L∈0,∞(ii)
un⟶L=0, and there exists R>0, such that nRun⟶+∞;(iii)
un⟶L=+∞, and there exists S>0, such that n−Sun⟶0.
For the scale model (5) with Y=∑k=1NSak,bk,ck,0, assume Sa,b. Then, the sequence of filtered statistical scale models En (15) have the LAN property with speed n at each value θ∈Θ. With HL given in (95), the asymptotic Fisher information quantity has the following expression:(97)Iθ0=ILθ2,IL=∫y2HL′2yHLydy−1.Remark 4.
(i)
Let us briefly explain the nature of the assumption in Theorem3. In [17], conditions of type Sa,b allowed us to show that Yti is distributed as an α-stable variable mixed on the skewness and scale parameters by other processes. More precisely, for all t>0, we have these identities in distribution: for all α in the interior of the domain D (given in Sa,b) and for all t>0, there exist a r.v. βt and γti, such that(98)Yti=dS1α,βt,γti,0=dγti1/αS1α,βt,1,0.The processesβ and γi are(99)βt≤B,Ctα/aiZt≤γti≤Dtα/aiZt,WhereZt=∑k=1nStak/α,1,1,0 is a sum of independent standard stable subordinators, and the nonnegative numbers C<D depend only on α,a1,b1,c1,…,aN,bN,cN. In the case where bkK¯ak is a constant for all k=1,…,N, then βt=b1K¯a1, γti is distributed as a normalized sum of independent stable subordinators, and the converge in the following distribution holds.(100)βt,γti⟶biK¯aiK¯α,S1ai/α,1,1,0,ifi=1andt⟶∞,or ifi=Nandt⟶0.Furthermore, notice that the assumptionSa,b is satisfied in the symmetrical cases(ii)
In general, whenβ,γ,δ are r.v.’s lying in the set of admissible parameters and ℱ is the σ-field generated by them, we gave in [17] a structure of the ℱ-conditional Lévy process to Srα,β,γ,δr≥0. See also [13] for the notion of the ℱ-conditional Lévy process. If β is deterministic, S1α,β,γ,0 is simply distributed as γ1/αS1α,β,1,0, and with our construction, we allow the same identity even if β is random and correlated with γ. We also considered in [17] the densities of some families of mixed stable variables S1α,βt,γt,0t∈T and gave several examples when these densities and their derivatives behave like the proper stable densities and this uniformly in t∈T.(iii)
It is also possible to state a version of Theorem3 with stable processes with drifts. It is enough to strengthen the conditions on the asymptotic as done in Theorem 2. We consider that Theorem 3 is far from being exhaustive. It is produced in the aim of illustrating the difficulty of this case. If one wants to reduce the assumption Sa,b on the coefficients, then additional controls on the densities are needed.Before tackling the proof of Theorem3, we need the following result borrowed from [17].Theorem 4 (See [17]).
Controls of the densities of some mixed stable variablesS1α,βr,γr,0r∈ℛ. Let γrr≥0 be a pure jump subordinator characterized by(102)Ee−λγ1=exp∫0,∞e−λx−1νdx,λ≥0.
Assumeνx=νx,∞=x−aLx, where a∈0,1, and L is a slowly varying function. For r,x>0, define(103)vr≔supt>0:νt>1r,νrx≔rνxvr,∞,γ¯r≔γrvr.(a)
IfL is slowly varying at infinity and r⟶∞ or if L is slowly varying at zero, then(104)νrx⟶1xa,x>0,γ¯r⟶dS1a,1,Γ1−a/a,0,asr⟶0+.(b)
Moreover, assume that there exist0<c≤a≤d<1 and K≥1, such that(105)yxa−c≤LyLx≤Kyxa−d,0<y<x.LetR′>R>0 and γrpt∈Rp denote one of these families: R1=R4=0,R,R2=R,R′,R3=R,∞, and γr1=γr2=γr, γr3=γ¯r with L slowly varying at ∞, and γr4=γ¯r, with L slowly varying at 0. Let α∈0,2 and βrr≥0 be any family of r.v.’s, such that supr≥0βr≤B, for some B∈0,1, (B=0, if α=1). Then, the p.d.f. Grp of the mixed stable variables S1α,βr,γrp are infinitely differentiable and satisfy the following: for all k∈ℕ and all p, we have(106)0<liminfx⟶∞infr∈Rpx1+αdGrpx,limsupx⟶∞supr∈Rpx1+αcGrpx<∞,limsupx⟶∞supr∈Rp,x∈ℝxkGrpkxGrpx<∞.For allX>0,p≠1, we have(c)
Ifγr′r≥0 is a family of r.v.’s, such that γr/K≤γr′≤Kγr for some K>1 and all r≥0, then the controls (106) and (17) remain true for the p.d.f.’s obtained by replacing γr,γ¯r by γr′,γr′/vr, where vr is given by (103).Remark 5.
Ifγ is a pure jump a-stable subordinator, then νrx=x−a, and the scaling property gives γ¯r=dS1a,1,Γ1−a/a. Furthermore, if βr=β,r>0, is deterministic, then S1α,βr,γ¯r=dS1αa,β′,γ′, for some β′∈−1,1,γ′>0. The estimates (106) and (17) are an immediate consequence of the behavior of the stable densities given in the Proof of Theorem 2.Example 1.
The following are examples of processes satisfying the conditions of Theorem4. Let 0<b,b1,…,bN<1, and c,c1,…,cN>0. Let γ1,γ2 be pure jump subordinators whose Lévy measures are, respectively, equal to(108)ν1dx=∑k=1Ncixbk+11lx>0dx,ν2dx=cxb+11lx>0dx+δ1dx.
The processγ1 is the sum of independent stable subordinators, and γ2 is the independent sum of a stable subordinator and a standard Poisson process. With b∨=maxbi and b∧=minbi, notice that(109)ν1x=L1∨xxb∨=L1∧xxb∧,ν2x=L2xxb,where L1∨,L2 are slowly varying at 0, L1 is slowly varying at ∞, and(110)0<y<x⟹yxb∨−b∧≤L1∨yL1∨x≤1,1≤L1∧yL1∧x≤yxb∧−b∨,yxb/2≤L2yL2x≤2.The following result is a consequence of Theorem4.Corollary 2.
AssumeSa,b. Let Hi,t be the p.d.f of Yti, i=1,N,t>0. Let T>0,T1=T,∞,TN=0,T, and(111)Fi,tkz=Hi,tkzHi,tz,k∈ℕ,z∈ℝ.Then, for every nonnegative integers, we have(112)0≤r≤k,0<ρ<1−1a1+1⟹supt∈TizrFi,tkzsHi,tz21−ρis integrable.Proof.
By Remark5, notice that the Lévy measure of subordinator Z in (99) has the required conditions. According to Theorem 4 (c), these conditions imply that the family H1,tt∈T1 behaves like Gr3r∈R3 and that HN,tt∈TN behaves like Gr4r∈R4 with c=a1/α and d=aN/α.Proof of Theorem3.
We will check the assumptions of Corollary1, which consist in the convergence of the integrals(113)In=∫1+zHi,un′Hi,unz2Hi,unzdz,Jnρ=γun2ρ−1/α∫1+zHi,un′Hi,unz2+2+4zHi,un′Hi,unz+z2Hi,un″Hi,unz2Hi,unz21−ρdz.
The latter is guaranteed by Corollary2, since the functions Fi,tk satisfy(114)supn∈ℕzrFi,unksHi,un21−ρ∈L1dz,for s∈0,1,2,3,4,r∈0,1,…,k,0≤k≤2, and ρ≤1−1/a1+1. The rest is obtained by reproducing the Proof of Theorem 2.Remark 6.
Notice that the main argument for proving Theorem3 is the behavior of the densities uniformly in time. Theorem 4 provides many other examples. For example, with the same proof as in Theorem 3, one could state a version with stable processes time changed by any independent “nice” subordinator. The time change process could be the sum of a stable subordinator and a Poisson process (Remark 5). Finally, it appears that more investigation concerning the behavior in small time of p.d.f.’s of Lévy processes attracted by stable processes is crucial for statistical purposes.


## 4.1. The Scale Models Associated with Stable Processes Have the LAN Property

The following result is a first application of Corollary1.Theorem 2.
For the scale model (5), assume Y=Sα,β,γ,δ is a stable process, such that δ=0 is null if α<1 and β=1. Recall δt,α is defined in (12). Let the path un satisfy ln=γun−1/αδun,α⟶l∈ℝ and one of the following conditions:(i)
un⟶L∈0,∞(ii)
un⟶L=0 and nRun⟶∞, for some R>0(iii)
un⟶L=+∞ and n−Sun⟶0, for some S>0
Then, the sequence of sequence of filtered statistical scale modelsEn (15) have the LAN property with speed n at each θ∈Θ. The asymptotic Fisher information quantity is given by(75)Iθ=1θ2∫y+l2G′y2Gydy−1,where G is the p.d.f. of S1α,β,1,0.Remark 3.
Ifα∈0,1, then support of the law of S1α,β,γ,δ is δ,∞ (respectively, −∞,δ) if β=1 (respectively, β=−1); otherwise, the support is whole ℝ. The assumption δ=0 ensures that the support of the law of S1α,β,γ,δ satisfies the condition (C0) of Corollary 1. We are aware that the cases nun⟶0andun⟶∞ are not very realistic by the statistical point of view. The most interesting cases are as follows:(a)
un=u∈0,∞, then L=u, and the assumptions of Theorem 2 are satisfied. This is an essentially trivial result because we treat then a regular i.i.d. model.(b)
un=u/n,u∈0,∞, then L=0, and the assumptions of Theorem 2 are satisfied if and only if one of the following holds:(i)
α<1,δ=0, and then, l=0;(ii)
α=1,β=0, and then, l=δ/γ;(iii)
α>1, and then, l=0.Proof.
of Theorem2. (1a) Recall that hn denotes the infinitely differentiable p.d.f of Sunα,β,γ,δ. Due to the scaling property (12), we have the following representations: for all k∈ℕ and x∈supporthn,(76)hnkx=1γunk+1/αG1α,β,1,0kx−δun,αγun1/α,where(77)Supporthn=ℝ+,if0<α<1,β=1,ℝ−,if0<α<1,β=−1,ℝ,otherwise.
(1b) In [14, 17], we provided several properties of the p.d.f. G=G1α,β,1,0. For instance, there exist positive constants A,B,C,D that depend explicitly on α,β, such that(78)Gx∼0+respectively,∼0−ξx,ifβ=1respectively,−1and0<α<1,∼+∞respectively,∼−∞χx,ifβ≠−1respectively,1and0<α<2,∼+∞respectively,∼−∞ηx,ifβ=−1respectively,1andα=1,∼+∞respectively,∼−∞ξx,ifβ=−1respectively,1and1<α≤2.where(79)χx≔Dxα+1,ξx≔Bx2−α/2α−1e−Axα/α−1,ηx=Cexp−eπx/2+πx4,and fx∼lgx means limx⟶lfx/gx=1. Furthermore, with the convention 0/0=0, the functions(80)Fkx≔GkxGx,k∈ℕ,are continuous on supporthn, given by (77), and there exist positive numbers a,b,c, depending explicitly on α and β and k, such that(81)Fkx∼0+respectively,∼0−axk/α−1,ifβ=1respectively,−1and0<α<1,∼+∞respectively,∼−∞bx−k,ifβ≠−1respectively,1and0<α<2,∼+∞respectively,∼−∞cexpkπx2,ifβ=−1respectively,1andα=1,∼+∞respectively,∼−∞axk/α−1,ifβ=−1respectively,1and1<α≤2.
From the last equivalences, we see that for any nonnegative integers, we have the implication(82)0≤r≤k⟹limx⟶∞x21−ρ1+αxrFksG21−ρ∈0,∞.
Thus, since0<ρ<1−1/21+α⇔21−ρ1+α>1, then(83)0≤r≤k,0<ρ<1−1α+1⟹x↦xrFkxsGx21−ρ∈L1dx.
(2) We need to verify the assumptions of Corollary1, i.e., to check (C1):(84)In=∫1+xhn′hnx2hnxdx⟶I=∫y+l2G′2Gydy−1∈0,∞,asn⟶∞,and also (C2); there exists a∈0,1/2, such that for ρ∈1/2,a,1−a, one has(85)1nJnρ=1n∫jn2hn2ρxdx⟶0,jnx=1+xhn′hnx2+2+4xhn′hnx+x2hn″hnxhnx.
(3) The scaling property (76) and the corresponding change of variables give the following representation of the quantity In in (84):(86)In=∫1+y+lnG′yGy2Gydy.
Using (82) and the fact that ln⟶l∈ℝ, we obtain(87)x↦supn∈ℕ1+y+lnG′Gy2Gy∈L1dy,limn⟶∞In=∫1+y+lG′Gy2Gydy.
Developing the last expression, integrating by parts and using the fact thatGy and yGy both tend to 0 as y goes to each endpoint of the support (77), we recover (84).
(4) Again, by the change of variables corresponding to (76) and by the representation (85), one has(88)Jnρ=γun2ρ−1/α∫1+y+lnG′Gy2+2+4y+lnG′Gy+y+ln2G″Gy2Gy21−ρdy.
Thanks to (83) and to the fact that ln⟶l∈ℝ, and it is clear that if ρ∈0,1−1/2α+1, then(89)supn∈ℕ1+y+lnG′Gy2+2+4y+lnG′Gy+y+ln2G″Gy2Gy21−ρis integrable. Let ϵ≔1/2−1/2α+1. To prove the convergence (85), it is enough to have(90)ρ∈0,ϵ+12,(91)limx⟶∞un2ρ−1/αn=0.
(5) Now, we distinguish between the values ofL=limn⟶∞un.(a)
IfL=0 and if ρ≥1/2, then (91) is true. By assumptions, we have 1/un≤nR, for large n. If ρ<1/2, then we have (91) as soon as R1−2ρ/α<1, which is equivalent to ρ>1/2−α/2R. Thus, we only have to choose ϵ′=ϵ∧α/4R and a=1/2−ϵ′ to get (90) to obtain (91) for ρ∈1/2,a,1−a.(b)
IfL∈0,∞, then (91) is always true.(c)
IfL=+∞ and if ρ≤1/2, then (91) is true. As in (5a), we have un≤nS for large n. If ρ>1/2 then we have (91) as soon as S2ρ−1/α<1, which is equivalent to ρ<1/2+α/2S. Thus, we only need to choose ϵ″=ϵ∧α/4Sanda=1/2−ϵ″ to obtain (90) and (91) for ρ∈1/2,a,1−a.


## 4.2. The Scale Models Associated with the Sum of Independent Stables Processes Have the LAN Property

This subsection gives a second example which also generalizes the previous one and achieves the situation (13). Define K¯α=1 if α≤1 and K¯α=α−2/α and assume the following restrictions on the skewness parameters:(92)Sa,b: we haveNindependent stable processesSak,bk,ck,0,such that the parameters satisfyaa1<a2<⋯<aN<2andD=∩k=1Nak,21+bkK¯ak≠∅;bbk=0ifak=1andB=maxbkK¯akK¯aN,1≤k≤N<1.LetY=∑k=1NSak,bk,ck,0 and Yi be the processes defined by(93)Yti=Ytt1/ai,i=1ori=Nand,t>0.The processesYi satisfy the asymptotic scaling property (72). Denote by Ht the p.d.f of Yt,(94)Ht=Gta1,b1,c1,0∗⋯∗GtaN,bN,cN,0,and by Hi,t, the one of Yti. Then, hn=Hun satisfies(95)hnkx=unci−k+1/αiHi,unkunci−1/aix.In [14, 17], we showed that the following functions H0 and H∞ given by(96)HL≔limt⟶∞H1,tx=G1a1,b1,1,0x,ifL=+∞,limt⟶0+HN,tx=G1aN,bN,1,0x,ifL=0,are well defined, and the above convergence holds uniformly in x∈supportG1ai,bi,1,0 and still hold for the successive derivatives. As one can guess, we are going to exploit identity (95) and state the following result.Theorem 3.
Letunn be a sequence satisfying one of the following conditions:(i)
un⟶L∈0,∞(ii)
un⟶L=0, and there exists R>0, such that nRun⟶+∞;(iii)
un⟶L=+∞, and there exists S>0, such that n−Sun⟶0.
For the scale model (5) with Y=∑k=1NSak,bk,ck,0, assume Sa,b. Then, the sequence of filtered statistical scale models En (15) have the LAN property with speed n at each value θ∈Θ. With HL given in (95), the asymptotic Fisher information quantity has the following expression:(97)Iθ0=ILθ2,IL=∫y2HL′2yHLydy−1.Remark 4.
(i)
Let us briefly explain the nature of the assumption in Theorem3. In [17], conditions of type Sa,b allowed us to show that Yti is distributed as an α-stable variable mixed on the skewness and scale parameters by other processes. More precisely, for all t>0, we have these identities in distribution: for all α in the interior of the domain D (given in Sa,b) and for all t>0, there exist a r.v. βt and γti, such that(98)Yti=dS1α,βt,γti,0=dγti1/αS1α,βt,1,0.The processesβ and γi are(99)βt≤B,Ctα/aiZt≤γti≤Dtα/aiZt,WhereZt=∑k=1nStak/α,1,1,0 is a sum of independent standard stable subordinators, and the nonnegative numbers C<D depend only on α,a1,b1,c1,…,aN,bN,cN. In the case where bkK¯ak is a constant for all k=1,…,N, then βt=b1K¯a1, γti is distributed as a normalized sum of independent stable subordinators, and the converge in the following distribution holds.(100)βt,γti⟶biK¯aiK¯α,S1ai/α,1,1,0,ifi=1andt⟶∞,or ifi=Nandt⟶0.Furthermore, notice that the assumptionSa,b is satisfied in the symmetrical cases(ii)
In general, whenβ,γ,δ are r.v.’s lying in the set of admissible parameters and ℱ is the σ-field generated by them, we gave in [17] a structure of the ℱ-conditional Lévy process to Srα,β,γ,δr≥0. See also [13] for the notion of the ℱ-conditional Lévy process. If β is deterministic, S1α,β,γ,0 is simply distributed as γ1/αS1α,β,1,0, and with our construction, we allow the same identity even if β is random and correlated with γ. We also considered in [17] the densities of some families of mixed stable variables S1α,βt,γt,0t∈T and gave several examples when these densities and their derivatives behave like the proper stable densities and this uniformly in t∈T.(iii)
It is also possible to state a version of Theorem3 with stable processes with drifts. It is enough to strengthen the conditions on the asymptotic as done in Theorem 2. We consider that Theorem 3 is far from being exhaustive. It is produced in the aim of illustrating the difficulty of this case. If one wants to reduce the assumption Sa,b on the coefficients, then additional controls on the densities are needed.Before tackling the proof of Theorem3, we need the following result borrowed from [17].Theorem 4 (See [17]).
Controls of the densities of some mixed stable variablesS1α,βr,γr,0r∈ℛ. Let γrr≥0 be a pure jump subordinator characterized by(102)Ee−λγ1=exp∫0,∞e−λx−1νdx,λ≥0.
Assumeνx=νx,∞=x−aLx, where a∈0,1, and L is a slowly varying function. For r,x>0, define(103)vr≔supt>0:νt>1r,νrx≔rνxvr,∞,γ¯r≔γrvr.(a)
IfL is slowly varying at infinity and r⟶∞ or if L is slowly varying at zero, then(104)νrx⟶1xa,x>0,γ¯r⟶dS1a,1,Γ1−a/a,0,asr⟶0+.(b)
Moreover, assume that there exist0<c≤a≤d<1 and K≥1, such that(105)yxa−c≤LyLx≤Kyxa−d,0<y<x.LetR′>R>0 and γrpt∈Rp denote one of these families: R1=R4=0,R,R2=R,R′,R3=R,∞, and γr1=γr2=γr, γr3=γ¯r with L slowly varying at ∞, and γr4=γ¯r, with L slowly varying at 0. Let α∈0,2 and βrr≥0 be any family of r.v.’s, such that supr≥0βr≤B, for some B∈0,1, (B=0, if α=1). Then, the p.d.f. Grp of the mixed stable variables S1α,βr,γrp are infinitely differentiable and satisfy the following: for all k∈ℕ and all p, we have(106)0<liminfx⟶∞infr∈Rpx1+αdGrpx,limsupx⟶∞supr∈Rpx1+αcGrpx<∞,limsupx⟶∞supr∈Rp,x∈ℝxkGrpkxGrpx<∞.For allX>0,p≠1, we have(c)
Ifγr′r≥0 is a family of r.v.’s, such that γr/K≤γr′≤Kγr for some K>1 and all r≥0, then the controls (106) and (17) remain true for the p.d.f.’s obtained by replacing γr,γ¯r by γr′,γr′/vr, where vr is given by (103).Remark 5.
Ifγ is a pure jump a-stable subordinator, then νrx=x−a, and the scaling property gives γ¯r=dS1a,1,Γ1−a/a. Furthermore, if βr=β,r>0, is deterministic, then S1α,βr,γ¯r=dS1αa,β′,γ′, for some β′∈−1,1,γ′>0. The estimates (106) and (17) are an immediate consequence of the behavior of the stable densities given in the Proof of Theorem 2.Example 1.
The following are examples of processes satisfying the conditions of Theorem4. Let 0<b,b1,…,bN<1, and c,c1,…,cN>0. Let γ1,γ2 be pure jump subordinators whose Lévy measures are, respectively, equal to(108)ν1dx=∑k=1Ncixbk+11lx>0dx,ν2dx=cxb+11lx>0dx+δ1dx.
The processγ1 is the sum of independent stable subordinators, and γ2 is the independent sum of a stable subordinator and a standard Poisson process. With b∨=maxbi and b∧=minbi, notice that(109)ν1x=L1∨xxb∨=L1∧xxb∧,ν2x=L2xxb,where L1∨,L2 are slowly varying at 0, L1 is slowly varying at ∞, and(110)0<y<x⟹yxb∨−b∧≤L1∨yL1∨x≤1,1≤L1∧yL1∧x≤yxb∧−b∨,yxb/2≤L2yL2x≤2.The following result is a consequence of Theorem4.Corollary 2.
AssumeSa,b. Let Hi,t be the p.d.f of Yti, i=1,N,t>0. Let T>0,T1=T,∞,TN=0,T, and(111)Fi,tkz=Hi,tkzHi,tz,k∈ℕ,z∈ℝ.Then, for every nonnegative integers, we have(112)0≤r≤k,0<ρ<1−1a1+1⟹supt∈TizrFi,tkzsHi,tz21−ρis integrable.Proof.
By Remark5, notice that the Lévy measure of subordinator Z in (99) has the required conditions. According to Theorem 4 (c), these conditions imply that the family H1,tt∈T1 behaves like Gr3r∈R3 and that HN,tt∈TN behaves like Gr4r∈R4 with c=a1/α and d=aN/α.Proof of Theorem3.
We will check the assumptions of Corollary1, which consist in the convergence of the integrals(113)In=∫1+zHi,un′Hi,unz2Hi,unzdz,Jnρ=γun2ρ−1/α∫1+zHi,un′Hi,unz2+2+4zHi,un′Hi,unz+z2Hi,un″Hi,unz2Hi,unz21−ρdz.
The latter is guaranteed by Corollary2, since the functions Fi,tk satisfy(114)supn∈ℕzrFi,unksHi,un21−ρ∈L1dz,for s∈0,1,2,3,4,r∈0,1,…,k,0≤k≤2, and ρ≤1−1/a1+1. The rest is obtained by reproducing the Proof of Theorem 2.Remark 6.
Notice that the main argument for proving Theorem3 is the behavior of the densities uniformly in time. Theorem 4 provides many other examples. For example, with the same proof as in Theorem 3, one could state a version with stable processes time changed by any independent “nice” subordinator. The time change process could be the sum of a stable subordinator and a Poisson process (Remark 5). Finally, it appears that more investigation concerning the behavior in small time of p.d.f.’s of Lévy processes attracted by stable processes is crucial for statistical purposes.


## 5. How to Build a LAN Model from Another LAN Model?

In this section, we investigate to which extent the choice of the asymptotic is crucial. Assume we start from a LAN model associated to the observations of a Lévy process X along a discretization scheme iun,1≤i≤n. Can we affirm that the model associated to the observations of X+X˜, where X˜ is another independent Lévy process, also enjoys the LAN property with the same discretization scheme? We need some preliminaries and two lemmas to answer the last question.Consider two independent Lévy processesY and N defined on some probability space Ω¯,F¯,ℙ¯ with values in the Skorohod space Ω=Dℝ+,ℝ (when the processes Y and N are seen as infinite-dimensional random variables). Assume that N is a nondrifted compound Poisson process with Lévy measure ν, and consider the process Y˜≔Y+N. Recall that the increment process Xn of X, observed along a scheme un, is defined in (14) by(115)Xjn=Xj+1un−Xjun,0≤j≤n−1.Forθ∈Θ, suppose we observe Xn=θYn,X˜n=θY˜n, and let(116)ℙθ=ℒawθY|ℙ¯,ℙ˜θ=ℒawθY˜|ℙ¯.The probability measureℙθn (respectively, ℙ˜θn and the scale models En (respectively, E˜n) correspond to X (respectively, X˜) as in (15) and (16).Recall that ifQ,Q′ are two probability measures on some sample space, then the total variation distance Q−Q′ is the quantity(117)Q−Q′=supϕ∈ΦEQϕ−EQ′ϕ,Φ=ϕ:Ω⟶−1,1,ϕmeasurable.Lecam’s lemma [2] is as follows:Lemma 1.
For every probability measuresQ,Q′,R,R′, we have the inequality(118)∫1∧dRdQ−dR′dQ′dQ+Q′≤Q−Q′+2R−R′+2Q−Q′R+R′1/2.We also have the following result.Lemma 2.
Iflimn⟶∞nun=0, then limn⟶∞supθ∈Θℙθn−ℙ˜θn=0.Proof.
Since(119)ℙ¯Nun∈dy=e−νℝun∑k=0∞unkk!ν∗kdy,ℙ¯Nun=0≥e−νℝun,and since N has stationary and independent increments, we have(120)ℙθn−ℙ˜θn=supϕ∈ΦEℙθnϕX−Eℙ˜θnϕX=supϕ∈ΦEℙθϕXn−Eℙ˜θϕXn=supϕ∈ΦEℙ¯ϕθYn−Eℙ¯ϕθY˜n=supϕ∈ΦEℙ¯ϕθYn−Eℙ¯ϕθYn+Nn=supϕ∈ΦEℙ¯ϕθYn−Eℙ¯ϕYn+Nn=supϕ∈ΦEℙ¯ϕYn−ϕYn+Nn1lNn≡/0≤2ℙ¯Nn≠0=21−ℙ¯Nn≡0=21−ℙ¯Nj+1un−Njun=0,∀0≤j≤n≤21−ℙ¯Nun=0n≤21−e−νℝnun.
It is now clear thatℙθn−ℙ˜θn goes to 0, uniformly in θ as nun⟶0.
Now, we are able to complete Far’s problem [18], which treats the case where Y is a Brownian motion and where the discretization path is un=1/n, i.e., limn⟶∞nun=1.Theorem 5.
Assumelimn⟶∞nun=0. If the scale model (5) En associated to the process X has the LAN property with speed n in a point θ0∈Θ, then so is the scale model E˜n associated to the process X˜.Proof.
of Theorem5. (1) Fix θ0∈Θ, and J is a finite subset of ℝ and ξ∈ℝ. We shall prove that the weak functional convergence (22) of the likelihood processes Zn,ηξη∈J of En yields the one of the likelihood processes Z˜n,ηξη∈J of E˜n. The expression of the likelihood processes is given by (21), and for more convenience, we denote them from now on by(121)Ztn=Zkn,ηξη∈J,Z˜tn=Z˜tn,ηξη∈J,Zt′=Zt′,ηξη∈J,t∈0,1.
We need to show the following convergence in laws:(122)ℒawZn|ℙξnn⟶ℒawZ′,ηξη∈J|ℙξ′⟹ℒawZ˜n|ℙ˜ξnn⟶ℒawZ′,ηξη∈J|ℙξ′,asn⟶∞,or equivalently, for every K-Lipschitz function f:Dℝ+,ℝJ⟶ℝ, bounded by a constant C>0, we need to show that(123)limn⟶∞EℙξnnfZn=Eℙξ′fZ′⟹limn⟶∞Eℙ˜ξnnfZ˜n=Eℙξ′fZ′.
(2) For such functionsf, we will control the difference:(124)EℙξnnfZn−Eℙ˜ξnnfZ˜n=EℙξnnfZn−Eℙ˜ξnnfZn+Eℙ˜ξnnfZn−Eℙ˜ξnnfZ˜n.
In virtue of Lemma2, we have(125)EℙξnnfZn−Eℙ˜ξnnfZ˜n≤Csupθ∈Θℙθn−ℙ˜θn⟶0,asn⟶∞.
For everyε>0, we have(126)Eℙ˜ξnnfZn−fZ˜n=Eℙ˜ξnnfZn−fZ˜n1lZn−Z˜n≤ε+Eℙ˜ξnnfZn−fZ˜n1lZn−Z˜n>εEℙ˜ξnnfZn−fZ˜n≤εK+2Cℙ˜ξnnZn−Z˜n>ε.
With representation (21), observe that Zn and Z˜n are a step process, time-dependent, up to ⌊nt⌋,t∈0,1. Then, denoting(127)τn=inf1≤j≤ns.t.Zjn−Z˜jn>ε,∀η∈J,ξ∈ℝ,and using Markov in equality, we obtain(128)ℙ˜ξnnZn−Z˜n>ε=ℙ˜ξnnZτnn−Z˜τnn>ε=ℙ˜ξnnZin−Z˜jn>ε,∀1≤j≤n≤1εEℙ˜ξnn1∧Zτnn−Z˜τnn.
Applying Lemma1 with R=ℙηnn,Q=ℙξnn,R′=ℙ˜ηnn,Q′=ℙ˜ξnn, we obtain(129)Eℙ˜ξnn1∧Zτnn−Z˜τnn≤2supθ∈Θ3ℙθn−ℙ˜θn+2ℙθn−ℙ˜θn1/2,and then, Lemma 2 gives(130)limn⟶∞supη∈J,ξ∈ℝℙ˜ξnnZn−Z˜n>ε=0.
The latter, together with (126), allows to conclude that(131)limn⟶∞EℙξnnfZn−fZ˜n=0.



---

*Source: 3143324-2021-12-29.xml*