When is it no longer possible to estimate a compound Poisson process?

We consider centered compound Poisson processes with ﬁnite variance, discretely observed over [0 , T ] and let the sampling rate ∆ = ∆ T → ∞ as T → ∞ . From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for ∆ T and we observe how the Fisher information degenerates. Then, we generalize these results to the class of compound Poisson processes. We establish a lower bound showing that consistent estimation is impossible when ∆ T grows faster than √ T . We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables.


Introduction 1.Motivation and statistical setting
Continuous diffusive models are often used for phenomena observed at large sampling rate, even though they present discontinuities or jumps at lower frequencies.For example in finance, asset prices or volumes change at discrete random times (see for instance Gerber and Shiu [7], Russell and Engle [18] or Guilbaud and Pham [8]), however it is common to use continuous diffusive processes to model them when the sampling rate is large (see e.g.Masoliver et al. [13], Önalan [16] or Hong and Satchell [9]).This opposition in the observations' behavior between small frequencies and large sampling rate is evoked in Cont and de Larrard [4]: "over time scales much larger than the interval between individual order book events, prices are observed to have diffusive dynamics and modeled as such."In physics the opposition between large scale diffusive behavior and point process at small scale is also popular (see e.g.Metzler and Klafter [14] or Uchaikin and Zolotarev [23]).The usual justification for using diffusive approximations is as follows.Suppose we have discrete observations of a centered pure jump process X observed at a sampling rate ∆ > 0, e.g. a centered compound Poisson process with finite variance, namely we observe X ∆ , . . ., X ⌊T ∆ −1 ⌋∆ . ( To lighten notation we set n T := ⌊T ∆ −1 ⌋, the number of observations.We make explicit the dependence in T since it is the asymptotic of the paper.If ∆ is large, between two observations of X many jumps occurred, the central limit theorem gives for every increments the approximation where W is a standard Wiener process and σ is positive.Hence, only the variance parameter σ 2 should be identifiable from (1).If X depends on more parameters their identifiability should be lost.Yet the use of diffusive approximations conceals the jump's dynamic observed at lower frequencies.The following questions naturally come across.
i) Is it possible to estimate the parameters characterizing X from (1)?
ii) Is the experiment generated by (1) asymptotically equivalent to a Gaussian experiment when ∆ = ∆ T → ∞ as T → ∞?
The asymptotic equivalence of a Poisson experiment with variable intensity has been studied in Brown et al. [3].Shevtsova [19] looks at the accuracy of Gaussian approximations for Poisson random sums.
The condition T /∆ T → ∞ ensures there are asymptotically infinitely many observations (⌊T ∆ −1 ⌋ → ∞ as T → ∞).A typical example of macroscopic regime is a sampling rate ∆ T of the order of T α as T → ∞ for α in (0, 1) as T → ∞.In this paper we restrain our study to homogeneous compound Poisson processes.A compound Poisson process X is defined as where R is a Poisson process of intensity λ and (ξ i ) are independent and identically distributed random variables independent of R. The process X is characterized by the the pair r = (λ, f ), where f is the probability law of ξ 1 .We denote by P the class of compound Poisson processes.

Main results
Investigating questions i) and ii) directly is difficult.Hence in Section 2 we first build and study a toy model: a compound Poisson process plus a drift that depends on a 2-dimensional parameter.This process does not belong to P. From this toy model, we identify two distinct macroscopic regimes, • A regime where ∆ goes to infinity faster than √ T , where the parameters cannot be consistently estimated from (1), providing a negative answer to i) (see Theorem 1 hereafter).
• A regime where ∆ goes to infinity slower than √ T , where the parameters can be estimated answering positively to i).However, optimal rates are much slower than usual parametric ones (see Proposition 1 hereafter).
From the study of the toy model, we derive a lower bound in Theorem 2. It identifies regimes in which consistent estimation of the law generating a process in P is impossible, leading to a negative answer to i).Theorem 3 gives an asymptotic equivalence result; according to the behavior of ∆ T with regard to T , the following occurs.
• The experiment generated by the observation of a process in P is asymptotically equivalent to a Gaussian experiment, answering positively to ii).
• Compound Poisson processes depending on a large number of parameters are not identifiable, providing a negative answer to i).The limit number of parameters beyond which consistent estimation is not possible is made explicit.
This paper is organized as follows, in Section 2 we construct and study our toy model.In Section 3 we establish the main Theorems 2 and 3. A discussion is proposed in Section 4. Finally, Section 5 is devoted to the proofs.

Information loss: A parametric example 2.1 Building up a parametric model
Consider the Lévy process Y defined by where R is Poisson process of intensity λ ∈ (0, ∞) independent of (ξ i ) i≥0 which are independent and exponentially distributed random variables with parameter β ∈ (0, ∞).
Due to the drift part, Y does not belong to P (unlike X).This model, known as the Cramér-Lundberg model, is used by insurance companies to model big claims of subscribers (see e.g.Embrechts et al. [6] or Miksoch [15]).Without the drift part, it is also used in Alexandersson [1] to model rainfall.
Suppose we observe ⌊T ∆ −1 ⌋ increments of Y , conditional on the event {R i∆ − R (i−1)∆ = 0}.Namely we observe Y over [0, S(T )∆] at a sampling rate ∆ > 0, where S(T ) is random and such that The following results on S(T ) can be easily checked.Since the probability of occurrence of a zero increment of X is e −λ∆ , we have if ∆ goes to infinity as a power of T .Moreover, S(T ) is negative binomial with parameters We introduce the family of experiments indexed by ∆ generated by the conditional observations (3) , where θ denotes the unknown parameter θ = (λ, β) ∈ Θ = (0, ∞) × (0, ∞) and P T,∆ θ the law of Y .

Remark 2. The natural experiment to work with is the experiment
But the law of Y is not dominated and the Fisher information in Y ∆ does not exist.Indeed the distribution of Y ∆ can be decomposed in where p ∆,θ is dominated by the Lebesgue measure but δ{− λ∆ β }, the mass concentrated at − λ∆ β , cannot be dominated over Θ. Removing null increments of the Poisson part gives the experiment Y ∆ , dominated by the Lebesgue measure, where the Fisher information exists.Since the probability of a null increments of X is e −λ∆ , which is negligible as ∆ → ∞, we show in Section 2.4 that the results established for Y hold also for Y : the experiments Y ∆ and Y ∆ are asymptotically equivalent.
The intuition of the problem is the following, as ξ 1 has finite variance, the central limit theorem applies for each increments and gives for i in J Thus each observation converges in law to a Gaussian random variable depending on one parameter: the parameter θ should no longer be identifiable when ∆ gets large, only the ratio λ/β 2 should be.

Study of the Fisher information
The increments of Y are independent and identically distributed, it follows that the Fisher information of Y ∆ satisfies where I 1,∆ (θ) is the Fisher information corresponding to one increment.It has no closed form expression but the following Proposition gives its asymptotic behavior.
and the eigenvalues of , denoted e 1,∆ T (θ) and e 2,∆ T (θ), satisfy Remark 3. The matrix I(θ) is the Fisher information of an experiment consisting in one variable of distribution N (0, 2λ/β 2 ).
From Proposition 1, whenever ∆ T goes to infinity faster than √ T the Fisher information degenerates to a rank 1 matrix: the second eigenvalue e 2,∆ T goes to 0 as ⌊T ∆ −1 T ⌋/∆ T ∼ T /∆ 2 T .Theorem 1 below shows that it is indeed not possible to build a consistent estimator of θ in those scales.Conversely, when ∆ T is slower than √ T , both eigenvalues of the Fisher information go to infinity.Since the experiment Y ∆ is regular we deduce that the parameter θ remains identifiable and that consistent estimators of θ do exist.This is surprising, even if each observation is close to a Gaussian variable depending on one parameter, the whole sample still permits to estimate consistently all unknown parameters.However the optimal rate of convergence, determined by the slowest eigenvalue e 2,∆ T (θ), is in (⌊T ∆ −1 T ⌋/∆ T ) 1/2 .It is much slower than usual parametric rates in ⌊T ∆ −1 T ⌋ 1/2 , the square root of the sample size.

A lower bound
In what follows • denotes a norm on R 2 .Define the diameter of a set A as Then, for all θ 0 ∈ Θ and δ > 0 there exists V δ (θ 0 ) ⊂ Θ a neighborhood of θ 0 such that diam(V δ (θ 0 )) ≤ δ and lim where the infimum is taken over all estimators.
From Theorem 1, there is no consistent estimator of θ when ∆ T grows rapidly to infinity, faster than √ T .This was expected as the Fisher information degenerates to a rank 1 matrix in those regimes (see Proposition 1).Notice that if Theorem 1 holds for every δ > 0, possibly small, it is not uniform in δ.It is not possible to apply it along a vanishing sequence of δ.

Generalization to the unconditional experiment
The asymptotic equivalence of Y ∆ and Y ∆ (defined in Remark 2) is an immediate consequence of the following Lemma.Lemma 1. Define the probability measures, where θ ∈ Σ, where Σ is a compact subset of R d , d ≥ 1, a n (θ) ∈ (0, 1), f n (θ, •) is a density absolutely continuous with respect to the Lebesgue measure and w n,θ is a probability measure.Consider the statistical experiments E n and G n generated by the independent and identically distributed observation of n random variables of density p n (θ, •) and q n (θ, •) respectively.If sup θ∈Σ a n (θ) = o( 1 n ), then E n and G n are asymptotically equivalent.Proof of Lemma 1 is given in Section 5. Since Y is a Lévy process, observations Y , and Y , are independent and identically distributed.The distribution of Y ∆ is where δ {−λ∆/β} is the measure concentrated at − λ∆ β and p ∆,θ is the density of Y ∆ absolutely continuous with respect to the Lebesgue.We consider macroscopic regimes such that ∆ = 0(T α ) for some α ∈ (0, 1), it follows that e

Identifiability loss for compound Poisson processes 3.1 A lower bound
In Section 2 we exhibit on a parametric example a regime where estimation is impossible.We generalize here Theorem 1 to the class of compound Poisson processes P whose norm Then, for all r 0 ∈ P, r 0 2,P , and δ > 0, there exists where the infimum is taken over all estimators.
It follows that if ∆ T is of the order of T α for α ∈ (1/2, 1) it is not possible to build a consistent estimator of (λ, f ) from (1) when f is unknown.

Remark 4. A compound Poisson process is a renewal reward process and a Lévy process. Thus, we immediately derive from
it is not possible to build consistent estimators of the law generating a renewal reward process or a Lévy process with jumps from (1).

Remark 5. The rate restriction T /∆
) is technical and might be weakened in T /∆ 2 T = O(1).Indeed we derive Theorem 2 from Theorem 1, which holds under the restriction T /∆ 2 T = O(1).To apply Theorem 1 in the present setting, we show that the experiment Y ∆ T introduced in Section 2 is asymptotically equivalent to an experiment generated by increments of a compound Poisson process.This asymptotic equivalence result imposes the constraint T /∆ 2 T = o((log(T /∆ T )) −1/4 ).

Building up asymptotically equivalent experiments
A parameter transformation function.Consider some density f θ with respect to the Lebesgue measure, centered with finite K first moments, K ∈ N. Let where R + denotes (0, ∞).Let γ > 0, consider the parameter transformation function In the sequel we consider X and Z two compound Poisson processes, X has intensity λ and compound density f θ and Z has intensity γλ and compound density f hγ (θ) .Namely f hγ (θ) is a density with respect to the Lebesgue measure such that To establish Theorem 3 below, we do not need to know f hγ (θ) , only to make sure of its existence and that it is in the Sobolev space W 1,1 (i.e.f hγ (θ) ∈ C 1 with f ′ hγ (θ) ∈ L 1 , see the class of densities (9) below).The existence of f hγ (θ) is an immediate consequence of the truncated Hamburger moment problem.A necessary and sufficient condition for (5) to have a solution is that the associated Hankel matrices are positive definite (see e.g.Athanassoulis and Gavriliadis [2] or Tagliani [21]).Since (0, m 2 , . . ., m K ) are the first moments of the density f θ , the associated Hankel matrices are positive definite.Then, the Hankel matrices of (0, m 2 γ , . . ., m K γ ), for any γ > 0, are positive definite as well and existence of a density f hγ (θ) , absolutely continuous with respect to the Lebesgue measure, is thus ensured.Note that the number of solutions is infinite.A lot of papers study methods to build explicit solutions of (5); for instance maximum entropy approaches (see e.g.Tagliani [21] or Sobezyk and Trȩbicki [20]), polynomial solutions (see e.g.Rodriguez and Seatzu [17]) or solutions based on kernel density functions (see e.g.Athanassoulis and Gavriliadis [2]).All these solutions are C 1 with integrable derivative, which ensures that f hγ (θ) can be chosen in W 1,1 .In practice if K is large, building f hγ (θ) is difficult and the shape of the solution highly depends on the construction method considered.Above references provide explicit examples.
Definition of the experiments.Consider also a Gaussian process W with quadratic variation λm 2 .We associate the parameter φ = (λ, m 2 ) in Σ 2 .Suppose X, Z and W are discretely observed at a sampling rate ∆ > 0 over [0, T ], namely Define the families of statistical experiments indexed by ∆ where P T,∆ θ denotes the law of ( 6), Q T,∆ θ the law of ( 7) and D T,∆ φ the law of (8).

Statement of the result
Define the subclass of densities where F(R) is the class of densities with respect to the Lebesgue measure and f * denotes the Fourier transform of f .The class F contains any density sufficiently regular.For instance all densities in the Sobolev space W 1,1 , i.e. densities with integrable derivatives.
), the experiments X ∆ T and Z ∆ T are asymptotically equivalent.

Moreover, if either one of the following holds
The assumptions of Theorem 3 focus on f θ only as it is possible to select f hγ (θ) solution of (5) satisfying the same assumptions (see Section 3.2.1).

Interpretation
Part 2 of Theorem 3 can be easily interpreted.It states that when ∆ T goes rapidly to infinity, the Gaussian approximation is valid.The increments of a compound Poisson process cannot be distinguished from the increments of a Brownian motion.Using a diffusive model even though the phenomena is per se discontinuous is justified in those regime (see e.g.Cont and de Larrard [4]).
Part 1 of Theorem 3 is more general since it holds regardless of the rate ∆ T .It should be interpreted as follows: for a given rate ∆ T , with regard to T, how many parameters are not identifiable?The response given by the theorem is if ∆ T is of the order of T α as T → ∞, α ∈ (0, 1), it is not possible to identify more than K α = ⌈ 2 α − 1⌉ moments of the compound law and the intensity.Indeed, it is possible to exhibit two different compound Poisson processes that cannot be distinguished from their discrete observation.Thus, compound laws characterized by their M ≥ K α first moments cannot be estimated consistently from observations (1).
The case K = 1, where parameter θ reduces to θ = λ, is not covered by Theorem 3. Indeed θ appears in the limit variance and is always identifiable.This case is studied for a particular discrete compound law in Duval and Hoffmann [5], where an efficient estimator of θ is given and the asymptotic equivalence with a Gaussian experiment is established for ∆ going rapidly to infinity, namely T /∆ 1+1/4 T = o((log(T /∆ T )) −1/4 ).This constraint is more restrictive than the one of Theorem 3 due to the discreteness of the compound law, a regularizing kernel is needed to prove the equivalence and imposes the condition.In the case K = 2, the parameter becomes θ = (λ, m 2 ), a particular example is studied in Section 2. Corroborating Theorem 3, Theorem 2 shows that it is not possible to estimate θ whenever T ∆ −2 T → 0 as T → ∞ since two parameters have to be estimated.

Discussion
Consequences and extensions.An immediate consequence of the results of the paper is that nonparametric estimation for compound Poisson processes is impossible when ∆ goes to infinity as a power of T , since it requires to estimate an infinite number of parameters (see Theorem 3).In this paper we did not investigate the existence and properties of consistent estimation procedures when they exist.From the example of Section 2, we may expect that such procedures exist but have optimal rates of convergence that deteriorate as the number of parameters increases.
A natural generalization of Theorem 3 would be to relax the constraint on the third moment of the compound law in | R x η f θ (x)dx| < ∞ for some η > 0, and more specifically for η ∈ (0, 2).This allows to exhibit at the limit a convergence to any stable process and not only to a Brownian motion (see for instance Kotulski [10] or Levy and Taqqu [12]).The stable limit law is parametric, then if the initial process depends on too many parameters questions i) and ii) (modifying the limit experiment accordingly) of Section 1.1 may also be extended.However, the methodology used in this paper highly rely on the hypothesis | R x 3 f θ (x)dx| < ∞ (see the proof of Theorem 3 and the use of Edgeworth expansions).Another generalization might be to add a long range dependence structure between the jump times or the jumps themselves that remains at the macroscopic limit.But our methodology uses heavily the Lévy structure of the process.
On the difficulty of giving identifiability results.Section 3 contains mostly negative results (see Theorems 2 and 3).Establishing positive results in the general case such as "parametric estimation is possible for K parameters if ∆ goes to infinity slower than T h(K) ", for some function h, is much more involved, even without specifying a rate of convergence.When ∆ goes to infinity, the law of each observation is asymptotically Gaussian and depends on one parameter, the asymptotic variance.If this variance is insufficient to recover the initial parameter, one has to study the limit experiment to derive identifiability and not just the law of one observation.The successful study of the example of Section 2 entirely relies on the fact that modified Bessel functions of the first kind appears in the density of the increments.Asymptotic expansions of such functions are known rendering possible the study of the limit Fisher information.Thus, the methodology adopted in Section 2 cannot be generalized to other cases.
The multivariate case.The results of the paper should apply also to multivariate compound Poisson models.In that case the limit distribution of the increments is a multivariate Gaussian variable, additional information might be extracted from the covariance structure between coordinates.
A particular model is worth mentioning.Consider the bidimentional compound Poisson model where we observe for each increment the number of events R i∆ − R (i−1)∆ and the value of the increment X i∆ −X (i−1)∆ .One may think this additional information may improve identifiability, it is not the case.Observing R enables to estimate the intensity of the Poisson process with high accuracy; indeed a sufficient statistics is the terminal value T ⌋∆ T and the maximum likelihood estimator R n T ∆ /T converges at the rate √ T which is much faster than √ n T !Consequently, it does improve identifiability when two parameters are to be estimated: the intensity and one jump parameter that appears in the variance.In that case, the example studied in Section 2 remains identifiable in all macroscopic regimes.Still, identifiability of the jump probability law remains to be studied.Intuitively, if f depends on parameters that cannot be fully identified from the asymptotic variance there might be a loss of identifiability: the variance of f is identifiable but maybe not its shape.Moreover, if the construction that permits to derive Theorem 3 part 1 becomes obsolete, the results of Theorem 3 part 2 still hold.

Preparation
The increments of Y are independent and identically distributed.Conditional on the presence of jumps, the density of Y ∆ + λ∆/β is where f β is the density of an exponentially distributed random variable with parameter β and ⋆ denotes the convolution product.It follows that f ⋆m β is the density of a gamma distribution.Then, for x ≥ 0 Let k ∈ N and introduce the function It is related to the modified Bessel function of the first kind I k as follows where .
Rewriting P ∆ [f β ] and adding the drift part we get the density p ∆,θ of Y ∆ , for x ≥ −λ∆/β Technical Lemmas Lemma 2. Let k ∈ N, the modified Bessel function of the first kind where the remainder depends on k and M .
We need to control the moments of Y ∆ and compute the first ones.For that we use relation (4) and the moments of Y ∆ derived from the Lévy-Kintchine formula The control of the moments of Y ∆ is given in Lemma 4 hereafter, which is a consequence of the following Lemma, whose proof can be found in the Appendix.
Lemma 3. Let K ∈ N, suppose X is a compound Poisson process whose compound law is centered and has moment up to order K. Then for ∆ large enough and , where C continuously depends on λ and the K first moments.
Proof of Lemma 4. The process Y is not in P, nevertheless a convex inequality leads to We apply Lemma 3 to the first term of the right hand part of the inequality.We control the second term using Faà di Bruno's formula; we compute the nth derivative of the Laplace transform of R ∆ − λ∆ at 0 as follows where F (t) = e λ∆(t−1) and G(t) = e t − t, which satisfy Applying Faà di Bruno's formula we get Let t = 0.All the terms corresponding to m 1 = 0 are null, we obtain for large enough ∆.The last inequality follows from the fact that, due to the constraint The constant C depends on λ.To conclude we control the moments of Y ∆ using (4) for ∆ large enough and where C continuously depends on θ.

Completion of the proof of Proposition 1
Since observations (3) are independent and identically distributed, the Fisher information satisfies where From ( 10) we derive g ′ k (x) = g k+1 (x).Straightforward computations lead to Finally equation ( 11), Lemma 2 applied with M = 8, Lemma 4 (with Remark 6) and the Taylor expansions around 0 of z → 1/(1 + z) up to order 4 in ∆ lead to Proposition 1.
Computations are made with Mathematica.
For the reader convenience, wherever there is no ambiguity, we drop the super and sub scripts as follows, , with the convention P ⊗1 = P.It follows that for any s > 0. The triangle inequality applied to (12) gives Noticing that for any s > 0 and P and Q some probabilities Markov's inequality and P − Q T V = |dP − dQ|, lead to Then, for all s > 0 where If x is bounded away from 1, Φ is strictly positive.In the remaining of the proof we choose h γ such that Define the function h γ : θ → h γ (θ) = (γ 2 λ, γβ) where γ = 1 is positive.First, Pinsker's inequality gives where K is the Kullback divergence and In view of Lemma 5, , and Then, if T /∆ 2 T → 0 as T → ∞, for large enough T there exits C 1 < 1 such that The inequality holds for any γ.
Remark 7. To bound the total variation norm in (14) we prefer the Kullback divergence over the Hellinger distance since the logarithm makes easier the manipulation of the density p θ,∆ (see Lemma 5).

Proof of Lemma 1
Both experiments E n and G n are dominated by ν n,θ (dx) = µ n,θ (dx) + dx, where µ n,θ is a dominating measure for w n,∆ , therefore to establish the asymptotic equivalence it is sufficient to show (see Le Cam and Yang [11]) where • T V denotes the total variation norm.Since each experiment is the n fold product of independent and identically distributed random variables 1 the result if X has density p n (θ, •) and Z has density q n (θ, •).The connection between the total variation norm and the L 1 norm leads to The condition sup θ∈Σ a n (θ) = o( 1 n ) completes the proof of Lemma 1.
Remark 8.The last inequality is an equality when µ n,θ and the Lebesgue measure are orthogonal.

Preliminary
The process Y defined in Section 2 is not in P, we build a compound Poisson process V close to Y in total variation norm.Keeping up with notation of Section 2, θ = (λ, β) ∈ Θ, where Θ is compact subset of (0, ∞) × (0, ∞), consider the process 1 For instance, by using the bound (see Tsybakov [22] pp.83-90) where N is a Poisson process of intensity 8 9 λ and independent of (ǫ i ) which are independent and identically distributed centered exponential variables with parameter 2 3 β.Their common density is Remark 9.The multiplicative constants 8 9 and 2 3 in front of λ and β ensure that Y ∆ defined by (2) and V ∆ have same moments of order 2 and 3.
its law.We have the following Lemma.
) and T /∆ T → ∞ as T → ∞.Then, for any compact set Θ ⊂ (0, ∞) × (0, ∞) Proof of Lemma 6 can be found in the Appendix.The steps of the proof follows the lines of the proof of Theorem 3 hereafter.

Completion of the proof of Theorem 2
Let r θ = (λ, f β ) defined from (20) and (21), for all r 0 ∈ F and δ > 0 where the neighborhood V δ (r 0 ) (resp.V δ (r θ 0 )) of r 0 (resp. It follows that for all r, r θ in V δ (r θ 0 ) we have r − r θ 2,P ≤ 2(δ + r θ 0 2,P ). ( The remainder of the proof is a consequence of Scheffé's theorem.Let F be a bounded function then for every measures P and It follows from ( 22) and ( 23) We conclude the proof with Lemma 6, Theorem 1 and taking limits.
where Re(z) designates the real part of z.Since (cos(ξ(x − y)) − 1)f (x)f (y) ≤ 0 for all x, y ∈ D we derive that cos(ξ(x − y)) = 1 for all x, y ∈ D. Since D is an interval of R we have ξ = 0.
Second, by Riemann-Lebesgue Lemma there exists A > 0 such that ∀|ξ| > A, |f * (ξ)| < 1/2.Since f has finite expectation ξ → |f * (ξ)| is continuous.It is continuous over the compact [ρ, A] and reaches its supremum, denoted S, which is by the first part of the proof strictly lower than 1.Finally, since f has finite variance f * ′′ (0) > 0 and for all ρ > 0 we have |f * (ρ)| < 1. Set a = S ∨ 1 2 ∨ |f * (ρ)|.Proof is now complete.To establish Theorem 3, we show that the total variation norm between the experiments vanishes, using the Lévy structure of the processes X, Z and W .The experiments X ∆ T , Z ∆ T and W ∆ T are dominated by the measure δ 0 (dx) + dx.Introduce where p ∆ T ,θ and q ∆ T ,hγ (θ) are the distributions of X ∆ T and Z ∆ T and p ∆ T ,θ and q ∆ T ,hγ (θ) are absolutely continuous with respect to the Lebesgue measure.For the reader convenience, in absence of ambiguity we drop the subscripts and set p := p ∆ T ,θ and q := q ∆ T ,hγ (θ) .
Proof of Theorem 3.1 We prove that for K ≥ 2, ∆ T satisfying the rate restriction and the condition sup θ∈Σ K | x 3 f θ (x)dx| < ∞ the experiments X ∆ T and Z ∆ T are asymptotically equivalent.They live on the same state space and are the ⌊T ∆ −1 T ⌋ fold product of independent and identically distributed random variables, therefore it is sufficient to show (see Section 5.3 the proof of Lemma 1) We have Where p and q are defined in ( 24) and ( 25 Set η = η T = κ ∆ T log(T /∆ T ), we claim that for κ 2 > 2λm, the terms I, II and III are o((T /∆ −1 T )) hence ( 27) and the result.Bounding terms II and III.For II we use that and that X ∆ T is a centered compound Poisson process whose compound law has finite variance, it follows that Let D ∼ N (0, λm 2 ), the triangle inequality gives We readily obtain We bound the second term using Edgeworth series, even if it means conditioning on the value of the Poisson process associated to X.By assumption, the compound law has finite moment of order 3, denoted m 3 , uniformly bounded over Σ K , we derive where C continuously depends on λ, m 2 and m 3 and which is o((T /∆ T ) −1 ) for κ 2 ≥ 2λm 2 .The term III is treated similarly as II, the parameter γ simplifies.We do not reproduce computations.Thus II and III have the right order, the choice of κ and the bounds on II and III is made independent of θ taking the supremum over the compact set Σ K .

Bounding term I. Plancherel theorem gives
where f * denotes the Fourier transform of f .The Fourier transforms are computed using ( 24) and ( 25) and the Lévy-Kintchine formula Then, 2πA can be upper bounded as follows where for ρ ≥ 0 Bounding term IV .Since f θ and f hγ (θ) have their K first moments finite, we get the following expansion for any bounded ξ for some bounded functions ξ ❀ α 1 (ξ) and ξ ❀ α 2 (ξ).It follows that IV is less than + 2ρ ∆ T (e −λ∆ T + e −γλ∆ T ).
We pick ρ such that ρ > 0 where the functions ∆ → P 2m (w, ∆), ∆ → Q 2m (w, ∆), ∆ → P 2m+1 (w, ∆) and ∆ → Q 2m+1 (w, ∆) are polynomials in ∆, the degree of Q 2m and Q 2m+1 is smaller than m and there exist C 1 functions (c 2m,j (•), c 2m+1,j (•), j = 1, . . ., m), continuously depending on λ, such that Assume that the property holds at rank m − 1, we have where P 2m have the desired property and from the property at rank m − 1 the degree of Q 2m is lower than m.Similar computations give the result for P 2m+1 and Q 2m+1 .We complete on the proof with (29), f * (0) = 1 and using that f is centered: where C 1 and C 2 continuously depend on λ.

Proof of Lemma 6
We adopt the same methodology as for the proof of Theorem 3. Computations are quite similar we do not develop all of them.Each experiment is the ⌊T ∆ −1 T ⌋-fold product of independent and identically distributed random variables the result is implied by (see Section 5.3 the proof of Lemma 1) uniformly over the compact set Θ. Let us further denote by p ∆ T ,θ and q ∆ T ,θ the densities of Y ∆ T and of V ∆ T respectively, which can be decomposed as follows p ∆ T ,θ (x) = e −λ∆ T δ 0 x − λ∆ T β + (1 − e −λ∆ T ) p ∆ T ,θ (x) (30) q ∆ T ,θ (x) = e − 8 9 λ∆ T δ 0 (x) + (1 − e − 8 9 λ∆ T ) q ∆ T ,θ (x) where p ∆ T ,θ and q ∆ T ,θ are absolutely continuous with respect to the Lebesgue measure.
For the reader convenience we set p := p ∆ T ,θ and q := q ∆ T ,θ .Then, we have that 2 P θ − Q θ T V equals T and w T,θ (dx) = δ {−λ∆ T /β} (dx).The experiments Y ∆ and Y ∆ are asymptotically equivalent and the results established for Y ∆ hold for Y ∆ .