The Modified Bidomain Model with Diffusive Inclusions Yves Coudière [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Gilbert | Visualization and quantification of whole rat heart laminar structure using high-spatial resolution contrast-enhanced MRI[END_REF][START_REF] Camelliti | Structural and functional characterisation of cardiac fibroblasts[END_REF][START_REF] Hall | Hall Textbook of Medical Physiology: Enhanced E-book[END_REF] , Anđela Davidović 1, 2, 3, 4 (andjela.davidovic@inria.fr) and Clair Poignard 1, 2, 3 1 INRIA Bordeaux Sud-Ouest, [START_REF] Gilbert | Visualization and quantification of whole rat heart laminar structure using high-spatial resolution contrast-enhanced MRI[END_REF] University Bordeaux, [START_REF] Camelliti | Structural and functional characterisation of cardiac fibroblasts[END_REF] Institut de Mathématiques de Bordeaux (IMB), [START_REF] Hall | Hall Textbook of Medical Physiology: Enhanced E-book[END_REF]

IHU-Liryc

The standard bidomain model •

Widely accepted model

• Very good for the healthy tissues • Derived from the microscopic problem • Assumptions used

• periodicity • uniformly distributed ECM • anisotropic conductivities

Drawbacks -Examples

• There are non-small regions where myocytes are not present

Modelling assumptions

• Periodic diffusive inclusions of size ε.

• Isotropic conductivity in the inclusions.

• No cells in the inclusions. Passive conductors.

• Standard transmission conditions.

Microscale model

ε Ω Ω B ε Ω D ε Σ ε ∂Ω Y Y D Y B Γ 1 • Bidomain model ∂ t h ε + g(v ε , h ε ) =0, in Ω B ε , ∂ t v ε + I ion (v ε , h ε ) =∇ • (σ i ε ∇u i ε ), in Ω B ε , ∂ t v ε + I ion (v ε , h ε ) = -∇ • (σ e ε ∇u e ε ), in Ω B ε .
• intra-and extra-cellular conductivities σ i ε and σ e ε

• unknown potentials u i ε and u e ε

• transmembrane potential v ε := u i ε -u e ε

• I ion (v, h) and g(v, h) define specific ionic model

• Diffusive inclusions ∇ • (σ d ε ∇u d ε ) = 0, in Ω D ε . • conductivity σ d ε • unknown potential u d ε • Interface σ i ε ∇u i ε • n Σ ε = 0, σ e ε ∇u e ε • n Σ ε = σ d ε ∇u d ε • n Σ ε , u e ε = u d ε ,                                    on Σ ε .

Method: Homogenisation

• Assume:

u i ε (t, x) = u i 0 (t, x) + u i 1 (t, x, x ε ) + • • • , u ε (t, x) = u 0 (t, x) + u 1 (t, x, x ε ) + • • • .
• Let ε → 0. Derive the homogenised problem and the cell problems. • According to [1], a priori estimates ensure the two-scale convergence

Limit macroscale model

• The limit problem in whole Ω:

∂ t h 0 + g(v 0 , h 0 ) =0, |Y B |(∂ t v 0 + I ion (v 0 , h 0 )) =∇ • (σ i ∇u i 0 ), |Y B |(∂ t v 0 + I ion (v 0 , h 0 )) = -∇ • (σ e ∇u 0 ). • New conductivities σi = σ i |Y B | + A i (σ i , w i ), σe = σ e |Y B | + σ d |Y D | + A(σ e , σ d , w)
where A i and A are matrices obtained by solving the cell problems on Y for unknown functions w i and w.

Remarks:

• The conductivities depend on the volume fraction and on the geometry of the diffusive part.

• If σ e = σ d then on the volume fraction only.

• If |Y D | = 0 it is the standard bidomain model. • If |Y B | = 0 it is only the diffusion model.

Numerical results

• FreeFem++ and gnuplot.

• Mitchell Schaeffer ionic model.

• 2D simulations on Ω rectangle, SBDF2 numerical scheme.

• Simple shapes of inclusions (circles and ellipses).

Conclusions

• Inclusions change the speed of propagation and depending on geometry, the anisotropy ratio. • Rigorous and practical way to link the structural disease to macroscopic conductivities.

Future work

• Use the late enhancement MRI for the volume fraction of the extracellular space. • Cells in the additional ECM. Boundary conditions.

Figure :

 : Figure:The patch of the cardiac tissue[4].

Figure :

 : Figure:The laminar structure of myocardium.[2]

Figure :

 : Figure: Fibroblast organization in sheep: normal ventricular myocardium (A), infarct border zone (B) and centre (C), 1 week after infarction. [3]

  inclusions. (c) Elliptical inclusions. Axes length ratio 1:4.Figure: Simulation of the standard bidomain with the "scar patch". σ d = 3, volume fraction 0.18, comparing inclusions geometries.

Table :

 : The values of the new conductivities depending on the diffusion inclusions [10 -1 S/m]. The first row: values for the standard bidomain model, without inclusions.

	geometry	vol frac σ d σ * i 11 σ * i 22 σ * e 11 σ * e 22
	-	-	-1.74 0.19 3.9 1.97
	circle	0.18	3 1.29 0.18 3.28 2.45
	ellipse (1:4) 0.18	3 0.31 0.19 0.75 2.61
	circle	0.2	3 1.26 0.17 3.29 2.53
	ellipse (1:1.5) 0.2	3 1.13 0.18 2.82 2.58
	ellipse (1:2) 0.2	3 0.86 0.18 2.09 2.64
	circle	0.4	3 1.07 0.15 3.42 3.53
	ellipse (1:1.5) 0.4	3 0.81 0.16 2.53 3.70
	circle	0.7	3 0.69 0.09 5.08 7.89


Yves Coudière

Anđela Davidović

Clair Poignard

Influence of periodic diffusive inclusions on the bidomain model

Introduction

We present a new mathematical model of the electric activity of the heart. The main drawback of the standard bidomain model is that it assumes the existence of excitable cells (myocytes) everywhere in the heart, while it is known that there exist non-small regions where non-excitable cells (fibroblasts and collagen) take place. The problems that we are trying to address with our model are:

• The laminar structure of the myocardium that shows the presence of collagen, especially between muscle layers. See Fig 1. [START_REF] Hooks | Laminar Arrangement of Ventricular Myocytes Influences Electrical Behavior of the Heart[END_REF][START_REF] Gilbert | Visualization and quantification of whole rat heart laminar structure using high-spatial resolution contrast-enhanced MRI[END_REF]. In the diseased tissue the diffusion space plays an important role. • The infarct border zone. After infarction in the heart some number of myocytes die and they are replaced by collagen and a few cells of fibroblasts. We consider that there are no cells of fibroblasts in the extracellular space.

I -Mesoscopic model

Modelling assumptions:

• Periodic distribution of added extracellular space • Extracellular space is a passive conductor

The domain Ω is split into two regions B ε and D ε :

• B ε represents the bidomain layer.

• intra-and extra-cellular conductivities σ i ε and σ e ε

• unknown potentials u i ε and u e ε

• define v ε := u i ε -u e ε
• D ε represents the diffusive inclusions.

• conductivity σ d ε • unknown potential u d ε • Σ ε = ∂B ε ∩ ∂D ε is the interface.
The bidomain model

∂ t v ε + cv ε =∇ • (σ i ε ∇u i ε ), in B ε , ∂ t v ε + cv ε = -∇ • (σ e ε ∇u e ε ), in B ε .
The diffusive inclusion

0 = -∇ • (σ d ε ∇u d ε ), in D ε .
The transmission conditions

σ i ε ∇u i ε • n Σ ε = 0, σ e ε ∇u e ε • n Σ ε = σ d ε ∇u d ε • n Σ ε , u e ε = u d ε ,                                      on Σ ε ,
The initial and the boundary conditions are given.

ε

Ω B ε D ε Σ ε ∂Ω Y Y D Y B Γ 1

II -Homogenisation

Heuristics: Redefine the problem in two scales and obtain the limit problem that depends only on the large scale variable. It gives a global, macroscopic behaviour of the unknown functions. Assume

σ i ε (x) = σ i        x ε        , σ e ε (x) = σ e        x ε        , σ d ε (x) = σ d        x ε        .
According to [START_REF] Allaire | Homogenisation and two-scale convergence[END_REF], a priori estimates ensure the twoscale convergence:

u i ε ։ u i 0 , (u e ε , u d ε ) ։ u 0 .

III -Macroscopic equations

The limit problem is still a bidomain model with updated conductivities:

∇ •    σi ∇u i 0    = ∂ t v 0 + cv 0 , in Ω, ∇ • ((σ e + σd ) ∇u 0 ) = -(∂ t v 0 + cv 0 ), in Ω, σi = σ i + 1 |Y B | A i , σe = σ e + 1 |Y B | A e , σd = |Y D | |Y B | σ d + 1 |Y B | A d ,
where A i , A e and A d are constant matrices that depend on the geometry of the unit cell.

Remark:

• The modified conductivities depend on the volume fraction of the diffusive part and on the geometry of the unit cell.

IV -Numerical verification on simple 2D geometries

Perspectives

• Apply the ionic current model.

• Use real data, e.g. the late enhancement MRI provides the volume fraction of the extracellular space.

Figure 1 :

 1 Figure1: The laminar structure of myocardium.[START_REF] Gilbert | Visualization and quantification of whole rat heart laminar structure using high-spatial resolution contrast-enhanced MRI[END_REF]

Figure 2 :

 2 Figure 2: organization in sheep: normal ventricular myocardium (A), infarct border zone (B) and centre (C), 1 week after infarction.[START_REF] Camelliti | Structural and functional characterisation of cardiac fibroblasts[END_REF]

•

  If |Y D | = 0 we obtain the standard bidomain model. If |Y B | = 0 we get only the diffusion model. • If σ e = σ d the modified extracellular conductivity will not depend on the geometry but only on the volume fraction.

Figure 3 :

 3 Figure 3: The time evolution for v ε = u i εu e ε , ε = 0.16 (left). The time evolution for v 0 = u i 0u 0 (right).


Sergueï Dachian
email: serguei.dachian@math.univ-bpclermont.fr

Lin Yang

On a Poissonian Change-Point Model with Variable Jump Size

Keywords: Poisson process, non-regularity, change-point, limiting likelihood ratio process, maximum likelihood estimator, Bayesian estimators Mathematics Subject Classification (2000) 62M05, 62M02

A model of Poissonian observation having a jump (change-point) in the intensity function is considered. Two cases are studied. The first one corresponds to the situation when the jump size converges to a non-zero limit, while in the second one the limit is zero. The limiting likelihood ratios in these two cases are quite different. In the first case, like in the case of a fixed jump size, the normalized likelihood ratio converges to a log Poisson process. In the second case, the normalized likelihood ratio converges to a log Wiener process, and so, the statistical problems of parameter estimation and hypotheses testing are asymptotically equivalent in this case to the well known problems of change-point estimation and testing for the model of a signal in white Gaussian noise. The properties of the maximum likelihood and Bayesian estimators, as well as those of the general likelihood ratio, Wald's and Bayesian tests are deduced form the convergence of normalized likelihood ratios. The convergence of the moments of the estimators is also established. The obtained theoretical results are illustrated by numerical simulations.

Introduction

In regular statistical experiments, the limit of the normalized likelihood ratio is always the same, because the families are LAN see, for example, [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] . In the case of non-regular statistical models for Poisson processes, there exists a large diversity of limiting likelihood ratio processes: change-point type models lead to a log Poisson process, "cusp" type singularities provide a log fBm process, while in the models with 0-type or ∞-type singularities the limit processes are more sophisticated see, respectively, [START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF][START_REF] Dachian | Estimation of cusp location by Poisson observation[END_REF][START_REF] Dachian | Estimation of the location of a 0-type or ∞-type singularity by Poisson observations[END_REF] . Note that in changepoint type models for diffusion processes, and particularly in the model of a discontinuous signal in white Gaussian noise (WGN), the limiting likelihood ratio is a log Wiener process see, for example, [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF][START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF] . It is interesting to investigate the relations between the different limit processes. This study was initiated in the recent works [START_REF] Dachian | On limiting likelihood ratio processes of some change-point type statistical models[END_REF][START_REF] Dachian | On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios[END_REF]. The present work is a part of this investigation, since we study a change-point model with variable jump size for Poissonian observations, and we obtain two different limits depending on the way the jump size is varying.

More precisely, we consider two cases. The first one corresponds to the situation when the jump size converges to a non-zero limit, while in the second one the limit is zero. The limiting likelihood ratios in these two cases are quite different. In the first case, as one could expect, the normalized likelihood ratio converges to a log Poisson process, just like the case of a fixed jump size. In the second case, the normalized likelihood ratio converges to a log Wiener process, that is, the statistical problems of parameter estimation and hypotheses testing are asymptotically equivalent to the well known problems of change-point estimation and testing for signal in WGN model. Let us note, that even if the latter result may seem unexpected, it is quite natural in the light of the recent work [START_REF] Dachian | On limiting likelihood ratio processes of some change-point type statistical models[END_REF] of one of the authors, where a relation between the log Poisson and the log Wiener limiting likelihood ratios was discovered.

Let us also mention that this situation is somewhat similar to what happens in the case of multi-phase regression models, where the limiting likelihood ratio is a log compound Poisson process in the case of a fixed jump size, while it is a log Wiener process in the case of a variable jump size converging to zero see, for example, [START_REF] Fujii | On weak convergence of the likelihood ratio process in multi-phase regression models[END_REF] and the references therein . Note also, that the recent work [START_REF] Dachian | On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios[END_REF] shades the light on the latter case, just as [START_REF] Dachian | On limiting likelihood ratio processes of some change-point type statistical models[END_REF] do in our case of Poissonian observations. Note finally, that we show not only the convergence of normalized likelihood ratios, but also the convergence of the moments of the estimators. This last convergence allows one, for example, to approximate the limiting mean square errors of the maximum likelihood and Bayesian estimators in the case of Poisson observations by the well known limiting mean square errors of these estimators calculated for signal in WGN model.

The paper is organized as follows. In Section 2 we describe the model of observations. In Section 3 we study the asymptotic behavior of the likelihood ratio. In Section 4, using the convergence of normalized likelihood ratio obtained in Section 3, we study the problem of parameter estimation. Similarly, in Section 5 we study the problem of hypothesis testing and illustrate the results by numerical simulations. Finally, Section 6 contains the proofs of all the lemmas.

2 Change-point model with variable jump size Suppose we observe n independent realizations X (n) j = X (n) j (t), t ∈ [0, τ] , j = 1, . . . , n, of an inhomogeneous Poisson process on the interval [0, τ] (the constant τ > 0 is supposed to be known) of intensity measure

Λ (n) ϑ A = A λ (n) ϑ (t) dt, A ∈ B [0, τ] ,
with intensity function λ

(n) ϑ , where ϑ ∈ Θ = (α, β ), 0 ≤ α < β ≤ τ,
is some unknown parameter. The observation will be denoted X

(n) = X (n) 1 , . . ., X (n) n
and the corresponding probability distribution will be denoted

P (n) ϑ .
Let us note that this model of observation is equivalent to observing a single realization on the interval [0, nτ] of an inhomogeneous Poisson process with the τ-periodic intensity function coinciding with λ

(n) ϑ on [0, τ].
The parameter ϑ corresponds to the location of a jump in the (elsewhere continuous) intensity function λ (n) ϑ . The size of the jump (depending on n) will be denoted r n and will be supposed converging to some r ∈ R. As we will see below, the behavior of our model depends on either one has r = 0 or r = 0 and is quite different in these two cases.

More precisely, we assume that the following conditions are satisfied.

(C1) The intensity function λ (C2) For all t ∈ [0, τ], there exist the lim n→+∞ ψ n (t) = ψ(t) > 0 and, moreover, this convergence is uniform with respect to t.

(C3) As n → +∞, the jump size r n converges to some r ∈ R, that is, r n → r. In the case r = 0, we also suppose that this convergence (r n → 0) is slower than n -1/2 , that is, n r 2 n → +∞.

(C4) The family of functions λ

(n) ϑ
n∈N,ϑ ∈Θ is uniformly strictly positive and uniformly bounded, that is, there exist some constants ℓ, L > 0 such that

ℓ ≤ λ (n) ϑ (t) ≤ L for all n ∈ N, ϑ ∈ Θ and t ∈ [0, τ].
Note that the conditions C1 -C3, together with the natural condition

r > -min t∈[0,τ] ψ(t), (1)
easily imply that the condition C4 holds for the family λ

(n) ϑ n≥n 0 ,ϑ ∈Θ with some n 0 ∈ N. So, in the asymptotic setting (n → +∞), the condition C4 can be replaced by ( 1), and we assume C4 instead of the latter only for convenience (as well as in order for our model to be well defined for all n ∈ N). Note also that in the case r = 0, the condition (1) is automatically satisfied.

An important particular case of this model is when only the jump size and not the regular part of λ

(n) ϑ depend on n. More precisely, the conditions C1 -C2 will be clearly met if we assume that the following condition is satisfied.

(C0) The intensity function λ (n) ϑ (t) can be written as λ (n) ϑ (t) = ψ(t) + r n 1 {t>ϑ } ,
where the function ψ is strictly positive and continuous on [0, τ].

Asymptotic behavior of the likelihood ratio

The likelihood of our model is given by see, for example, [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF]

L n ϑ , X (n) = exp n ∑ j=1 [0,τ] ln λ (n) ϑ (t) X (n) j (dt) -n τ 0 λ (n) ϑ (t) -1 dt = exp n ∑ j=1 ∑ i∈I (n) j ln λ (n) ϑ (t j,i ) -n τ 0 λ (n) ϑ (t) -1 dt , (2)
where t j,i , i ∈ I (n) j , are the jump times of the process X (n) j . Note that as function of ϑ , each λ (n) ϑ (t j,i ) is discontinuous (has a jump and is right continuous) at ϑ = t j,i . So, L n • , X (n) is a random process with càdlàg (continuous from the right and having finite limits from the left) trajectories.

We put ϕ n = 1 n in the case r = 0 and ϕ n = 1 nr 2 n in the case r = 0, and we introduce the normalized likelihood ratio

Z n,ϑ (u) = L n ϑ + uϕ n , X (n) L n ϑ , X (n) = exp n ∑ j=1 [0,τ] ln λ (n) ϑ +uϕ n (t) λ (n) ϑ (t) X (n) j (dt) -n τ 0 λ (n) ϑ +uϕ n (t) -λ (n) ϑ (t) dt = exp n ∑ j=1 ∑ i∈I (n) j ln λ (n) ϑ +uϕ n (t j,i ) λ (n) ϑ (t j,i ) -n τ 0 λ (n) ϑ +uϕ n (t) -λ (n) ϑ (t) dt , where u ∈ U n = ϕ -1 n (α -ϑ ), ϕ -1 n (β -ϑ ) .
Note that in both cases we have (by the condition C3 in the case r = 0) ϕ n → 0. Note also that if u > 0, we can rewrite Z n,ϑ (u) as

Z n,ϑ (u) = exp n ∑ j=1 (ϑ ,ϑ +uϕ n ] ln ψ n (t) ψ n (t) + r n X (n) j (dt) + n ϑ +uϕ n ϑ r n dt = exp n ∑ j=1 ∑ i ln ψ n (t j,i ) ψ n (t j,i ) + r n + u r γ n . (3)
In the last expression the inner sum is taken over the set i ∈ I

(n) j : ϑ < t j,i ≤ ϑ + uϕ n and we have γ = 1 in the case r = 0 and γ = -1 in the case r = 0.

Similarly, if u < 0, we have

Z n,ϑ (u) = exp n ∑ j=1 (ϑ +uϕ n ,ϑ ] ln ψ n (t) + r n ψ n (t) X (n) j (dt) -n ϑ ϑ +uϕ n r n dt = exp n ∑ j=1 ∑ i ln ψ n (t j,i ) + r n ψ n (t j,i ) + u r γ n ,
where the last sum is taken over the set i ∈ I

(n) j

: ϑ + uϕ n < t j,i ≤ ϑ and γ is as above. Note equally, that the process ln Z n,ϑ has independent increments. Indeed, its increments on disjoint intervals involve stochastic integrals (of a deterministic function with respect to Poisson processes) on disjoint intervals, and hence are independent. In other words, using the terminology of Strasser [START_REF] Strasser | Statistical experiments with independent increments[END_REF], our model is an "experiment with independent increments". Note also, that in this case the process Z n,ϑ as well as, for example, the process Z 1/2 n,ϑ is clearly a Markov process.

Note finally, that the trajectories of the process Z n,ϑ are càdlàg functions. Moreover, correctly extending these trajectories to the whole real line, one can consider that they belong to the Skorohod space D 0 (R). This space is defined as the space of functions f on R which do not have discontinuities of the second kind and which are vanishing at infinity, that is, such that lim u→±∞ f (u) = 0. We assume that all the functions f ∈ D 0 (R) are continuous from the right (are càdlàg).

Let us recall that the Skorohod metric on the space D 0 (R) is introduced by

d( f , g) = inf λ sup u∈R f (u) -g λ (u) + sup u∈R |u -λ (u)| ,
where the inf is taken over all strictly increasing continuous one-to-one mappings λ : R → R.

Let us also recall a criterion of weak convergence in D 0 (R). We put

∆ h ( f ) = sup u∈R sup u ′ ,u ′′ min f (u ′ ) -f (u) , f (u ′′ ) -f (u) + sup |u|>1/h | f (u)|,
where the inner sup is over all u ′ , u ′′ such that uh ≤ u ′ < u ≤ u ′′ < u + h. A criterion of weak convergence in D 0 (R) is given in the following lemma see [START_REF] Gihman | Introduction to the Theory of Random Processes[END_REF] for more details .

Lemma 1 Let z n,ϑ , n ∈ N, and z ϑ be random processes with realizations belonging to D 0 (R) with probability 1. If, as n → +∞, the finite dimensional distributions of z n,ϑ converge uniformly in ϑ ∈ K to the finite dimensional distributions of z ϑ , and if for any δ > 0

lim h→0 sup n∈N,ϑ ∈K P ∆ h (z n,ϑ ) > δ = 0, (4)
then, uniformly in ϑ ∈ K, the process z n,ϑ converges weakly in the space D 0 (R) to the process z ϑ .

Note that here and in the sequel K denotes an arbitrary compact in Θ .

The main objective of this section is the study of the asymptotic behavior in the sense of the weak convergence in the space D 0 (R) as n → ∞ of the above introduced normalized likelihood ratio Z n,ϑ . This behavior depends on either one has r = 0 or r = 0 and is quite different in these two cases, so the limit process must be introduced in a different manner in these two cases.

Case r = 0 limit process In the case r = 0, the limit process is a log Poisson type process and is introduced by

Z ϑ (u) =      exp ln ψ(ϑ ) ψ(ϑ )+r X + (u) + ru , if u ≥ 0, exp ln ψ(ϑ )+r ψ(ϑ ) X -(-u)-+ ru , if u < 0,
where X + and X -are independent Poisson processes on R + of constant intensities ψ(ϑ ) +r and ψ(ϑ ) respectively.

Let us note that Z ϑ (u) d = Z * ρ (-ru) with the constant ρ = ln ψ(ϑ ) ψ(ϑ )+r and the process Z * ρ defined by

Z * ρ (v) = exp ρY + (v) -v , if v ≥ 0, exp -ρY -(-v)--v , if v < 0,
where Y + and Y -are independent Poisson processes on R + of constant intensities 1 e ρ -1 and 1

1-e -ρ respectively. Note also that the process Z * ρ was recently studied in [START_REF] Dachian | On limiting likelihood ratio processes of some change-point type statistical models[END_REF] and that its trajectories (as well as those of the process Z ϑ ) almost surely belong to the space D 0 (R). More rigorously, in order to keep all the trajectories in the space D 0 (R), above we should rather have written

Z ϑ (u) d = Z * ρ (-ru)-in the case r > 0 .
Case r = 0 limit process In the case r = 0, the limit process is a log Wiener type process and is introduced by

Z ϑ (u) = exp ψ -1/2 (ϑ )W (u) - |u| 2ψ(ϑ ) , u ∈ R,
where W (u), u ∈ R, is a double-sided Brownian motion (Wiener process).

Let us note that Z ϑ (u)

d = Z * u/ψ(ϑ ) with the process Z * defined by Z * (v) = exp W (v) - |v| 2 , v ∈ R. (5)
Note also that the trajectories of the processes Z * and Z ϑ almost surely belong to the space C 0 (R) of continuous functions on R vanishing at infinity, and that C 0 (R) ⊂ D 0 (R). Now we can state the following theorem about the asymptotic behavior of the normalized likelihood ratio.

Theorem 1 Let the conditions C1 -C4 be fulfilled. Then, uniformly in ϑ ∈ K, the process Z n,ϑ converges weakly in the space D 0 (R) to the process Z ϑ .

Let us also remark, that sometimes it may be more convenient to use a slightly different rate for introducing the normalized likelihood ratio. More precisely, one can use the rate ϕ * n = 1 |r|n (rather than ϕ n = 1 n ) in the case r = 0, and the rate

ϕ * n = ψ(ϑ ) nr 2 n (rather than ϕ n = 1 nr 2 n ) in the case r = 0. That is, one can consider (instead of Z n,ϑ ) the normalized likelihood ratio Z * n,ϑ defined by Z * n,ϑ (v) = L n ϑ + v ϕ * n , X (n) L n ϑ , X (n) = Z n,ϑ c v
with c = 1/ |r| in the case r = 0, and c = ψ(θ ) in the case r = 0. Then, Theorem 1 will be clearly transformed to the following (equivalent) statement.

Theorem 2 Let the conditions C1 -C4 be fulfilled. Then, uniformly in ϑ ∈ K, the process Z * n,ϑ converges weakly in the space D 0 (R) to the process Z * ρ , in the case r < 0, the process Z ⋆ ρ defined by Z ⋆ ρ (v) = Z * ρ (-v)-, in the case r > 0, the process Z * , in the case r = 0.

The proof of Theorem 1 consist in checking the criterion of week convergence given in Lemma 1. For this, we follow the methods and ideas used in [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF]Chapters 5.3 and 5.4] and establish several lemmas (the proofs of the lemmas are in Section 6).

Lemma 2 Let the conditions C1 -C4 be fulfilled. Then the finite-dimensional distributions of the process Z n,ϑ converge to those of the process Z ϑ , and this convergence is uniform with respect to ϑ ∈ K.

Lemma 3 Let the conditions C1 -C4 be fulfilled. Then there exists a constant C > 0 such that

E (n) ϑ Z 1/2 n,ϑ (u 1 ) -Z 1/2 n,ϑ (u 2 ) 2 ≤ C |u 1 -u 2 | for all n ∈ N, u 1 , u 2 ∈ U n and ϑ ∈ K.
Lemma 4 Let the conditions C1 -C4 be fulfilled. Then there exists a constant k

* > 0 such that E (n) ϑ Z 1/2 n,ϑ (u) ≤ exp -k * |u| for all u ∈ U n , ϑ ∈ K and sufficiently large values of n (all n ∈ N in the case r = 0).
Final argument of the proof of Theorem 1 in the case r = 0 In this case, defining Z 1/2 n,ϑ ;a.c. to be the absolutely continuous component of the function Z 1/2 n,ϑ and, for p = 1, 2, denoting A p = A p (u, u + h) the event that Z n,ϑ has at least p jumps on the interval (u, u + h), we also have the following lemma.

Lemma 5 Let the conditions C1 -C4 be fulfilled with r = 0. Then the inequalities

E (n) ϑ Z 1/2 n,ϑ ;a.c. (u + h) -Z 1/2 n,ϑ ;a.c. (u) 2 ≤ Ch 2 , P (n) ϑ (A 1 ) ≤ D 1 h (6)
and

P (n) ϑ (A 2 ) ≤ D 2 h 2 (7)
hold with certain constants C, D 1 , D 2 > 0 (independent of n, ϑ , u and h).

Now, with the help of the above lemmas, we can finish the proof of Theorem 1 in the case r = 0 following the standard argument of [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF]Chapters 5.3 and 5.4]. More precisely, the weak convergence in D 0 (R) of the processes Z n,ϑ to the process Z ϑ follows from Theorem 5.4.2 of [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF], which is, in fact, contained in [START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] (without being formulated there). Note, that the conditions of this theorem are nothing but Lemmas 2, 4 and 5, and that its proof consist in verifying the condition (4).

Final argument of the proof of Theorem 1 in the case r = 0 In this case, it is not possible to establish a lemma similar to Lemma 5. In particular, the inequalities ( 6) and ( 7) do not hold, since in this case (in contrary to the case r = 0) the jumps are not becoming seldom. More precisely, as n → +∞, instead of having (on any finite interval) few "non-vanishing" jumps, one has more and more jumps which at the same time become smaller and smaller (which explains that the trajectories of the limiting likelihood ratio process in this case are continuous but nowhere differentiable functions). So, in order to finish the proof of Theorem 1 we use a different technique.

Since the increments of the process ln Z n,ϑ are independent, the convergence of its restrictions (and hence of those of Z n,ϑ ) on finite intervals [A, B] ⊂ R that is, convergence in the Skorohod space D [A, B] of functions on [A, B] without discontinuities of the second kind follows from Theorem 6.5.5 of Gihman and Skorohod [START_REF] Gihman | The Theory of Stochastic Processes I[END_REF], Lemma 2 and the following lemma.

Lemma 6 Let the conditions C1 -C4 be fulfilled with r = 0. Then for any ε > 0 we have

lim h→0 lim n→+∞ sup |u 1 -u 2 |<h P (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) > ε = 0. for all u 1 , u 2 ∈ U n and ϑ ∈ K.
Let us note, that taking a closer look on the proof of this lemma, one can see that we have even a stronger result: for any ε > 0 we have

lim h→0 lim n→+∞ sup ϑ ∈K sup |u 1 -u 2 |<h P (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) > ε = 0 for all u 1 , u 2 ∈ U n ,
which allow us to conclude that the convergence of the restrictions of the process Z n,ϑ on finite intervals [A, B] ⊂ R to those of the process Z ϑ is uniform with respect to ϑ ∈ K. Note also, that an alternative way to prove this convergence (instead of using Lemmas 2 and 6) is to study the characteristics of the processes and apply, for example, Theorem 7.3.4 of Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]. However, in our opinion, the proof given here gives more insight on the structure of the considered processes.

In order to conclude the proof of Theorem 1 applying the criterion of week convergence in D 0 (R) given in Lemma 1, we need to check the condition (4). Since we have already established the convergence of the restrictions on finite intervals [A, B] ⊂ R, it remains to control the second term of the modulus of continuity ∆ h (Z n,ϑ ) see, for example, [11, Chapters 5.3 and 5.4] . So, the last ingredient of the proof of Theorem 1 is the following estimate on the tails of the process Z n,ϑ .

Lemma 7 Let the conditions C1 -C4 be fulfilled with r = 0. Then there exist some constants b,C > 0 such that

P (n) ϑ sup |u|>D Z n,ϑ (u) > e -bD ≤ Ce -bD (8)
for all D ≥ 0, n ∈ N and ϑ ∈ K.

Parameter estimation

In this section we apply the convergence of normalized likelihood ratio obtained in Section 3 to study the problem of parameter estimation for our model of observations. In the case r = 0, the limiting likelihood ratio being the same as in the fixed jump size case, the properties of estimators are also the same see, for example, [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF][START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF] for more details . So, here we consider the case r = 0 only. Recall that as function of ϑ , the likelihood of our model given by ( 2) is discontinuous (has jumps). So, the maximum likelihood estimator ϑ n of ϑ is introduced through the equation max

L n ϑ n +, X (n) , L n ϑ n -, X (n) = sup ϑ ∈Θ L n ϑ , X (n) .
The Bayesian estimator ϑ n of ϑ for a given prior density p and for square loss is defined by

ϑ n = β α ϑ p(ϑ )L n ϑ , X (n) dϑ β α p(ϑ )L n ϑ , X (n) dϑ .
We are interested in the asymptotic properties of the maximum likelihood and Bayesian estimators of ϑ as n → +∞. To describe the properties of the estimators we need some additional notations.

We introduce the random variables ξ ϑ , ξ * , ζ ϑ and ζ * by the equations Theorem 3 Let the conditions C1 -C4 be fulfilled with r = 0. Then, for any ϑ 0 ∈ Θ , we have lim

Z ϑ (ξ ϑ ) = sup u∈R Z ϑ (u), Z * (ξ * ) = sup u∈R Z * (u), ζ ϑ = +∞ -∞ u Z ϑ (u) du +∞ -∞ Z ϑ (u) du and ζ * = +∞ -∞ u Z * (u) du +∞ -∞ Z * (u)
δ →0 lim n→+∞ inf ϑ n sup |ϑ -ϑ 0 |<δ ϕ -2 n E (n) ϑ (ϑ n -ϑ ) 2 ≥ Eζ 2 ϑ 0 = ψ 2 (ϑ 0 ) E(ζ * ) 2 ,
where the inf is taken over all possible estimators ϑ n of the parameter ϑ .

This theorem allows us to introduce the following definition.

Definition 1 Let the conditions C1 -C4 be fulfilled with r = 0. We say that an estimator

ϑ * n is asymptotically efficient if lim δ →0 lim n→+∞ sup |ϑ -ϑ 0 |<δ ϕ -2 n E (n) ϑ (ϑ * n -ϑ ) 2 = Eζ 2 ϑ 0 = ψ 2 (ϑ 0 ) E(ζ * ) 2
for all ϑ 0 ∈ Θ . Now, we can state the following two theorems giving the asymptotic properties of the maximum likelihood and Bayesian estimators.

Theorem 4 Let the conditions C1 -C4 be fulfilled with r = 0. Then the maximum likelihood estimator ϑ n satisfies uniformly on ϑ ∈ K the relations

P (n) ϑ -lim n→+∞ ϑ n = ϑ , L (n) ϑ ϕ -1 n ( ϑ n -ϑ ) ⇒ L (ξ ϑ ) = L ψ(ϑ )ξ * and lim n→+∞ E (n) ϑ ϕ -p n ϑ n -ϑ p = E |ξ ϑ | p = ψ p (ϑ ) E |ξ * | p for any p > 0.
In particular, the relative asymptotic efficiency of

ϑ n is E(ζ * ) 2 /E(ξ * ) 2 .
Theorem 5 Let the conditions C1 -C4 be fulfilled with r = 0. Then, for any continuous strictly positive density, the Bayesian estimator ϑ n satisfies uniformly on ϑ ∈ K the relations

P (n) ϑ -lim n→+∞ ϑ n = ϑ , L (n) ϑ ϕ -1 n ( ϑ n -ϑ ) ⇒ L (ζ ϑ ) = L ψ(ϑ )ζ * and lim n→+∞ E (n) ϑ ϕ -p n ϑ n -ϑ p = E |ζ ϑ | p = ψ p (ϑ ) E |ζ * | p for any p > 0.
In particular, ϑ n is asymptotically efficient.

Hypothesis testing

In this section we apply the convergence of normalized likelihood ratio obtained in Section 3 to study the problem of hypothesis testing for our model of observations. In the case r = 0, the limiting likelihood ratio being the same as in the fixed jump size case, the properties of test are also the same see [START_REF] Dachian | On hypotheses testing for Poisson processes[END_REF] for more details . So, here we consider the case r = 0 only.

We consider the same model of observation as above, with the only difference that now we suppose that θ

∈ Θ = [ϑ 1 , b), 0 < ϑ 1 < β ≤ τ.
We assume that the conditions (C1)-(C4) are fulfilled with r = 0 and we want to test the following two hypothesis:

H 1 : ϑ = ϑ 1 , H 2 : ϑ > ϑ 1 .
We define a (randomized) test φ n = φ n X (n) as the probability to accept the hypothesis H 2 . The size of the test is defined by E (n) ϑ 1 φ n X (n) , and its power function is given by

β (φ n , ϑ ) = E (n) ϑ φ n X (n) , ϑ > ϑ 1 .
As usually, we denote K ε the class of tests of asymptotic size ε ∈ [0, 1], that is,

K ε = φ n : lim n→+∞ E (n) ϑ 1 φ n X (n) = ε .
Our goal is to construct some tests belonging to this class and to compare them. The comparison of tests can be done by comparison of their power functions. It is known that for any reasonable test and for any fixed alternative the power function tends to 1. To avoid this difficulty, we use Pitman's approach and consider contiguous (or close) alternatives. More precisely, changing the variable ϑ = ϑ u

∆ = ϑ 1 + uϕ * n , where ϕ * n = ψ(ϑ 1 ) nr 2 n
, the initial problem of hypotheses testing can be replaced by the following one

H 1 : u = 0, H 2 : u > 0,
and the power function is now

β (φ n , u) = E (n) ϑ n φ n X (n) , u > 0.
The study is essentially based on the properties of the normalized likelihood ratio established above. Note that the limit of the normalized likelihood ratio at the point ϑ = ϑ 1 (under hypothesis H 1 ) is the following:

Z * n,ϑ 1 (v) = L n ϑ 1 + vϕ * n , X (n) L n ϑ 1 , X (n) ⇒ Z * (v), v ≥ 0,
where the process Z * is defined by [START_REF] Dachian | On hypotheses testing for Poisson processes[END_REF].

Under alternatives, we obtain

Z * n,ϑ 1 (v) = L n ϑ 1 + vϕ * n , X (n) L n ϑ 1 , X (n) = L n ϑ 1 , X (n) L n ϑ u , X (n) -1 L n ϑ 1 + vϕ * n , X (n) L n ϑ u , X (n) = L n ϑ u -uϕ * n , X (n) L n ϑ u , X (n) -1 L n ϑ u + (v -u)ϕ * n , X (n) L n ϑ u , X (n) ⇒ Z * (-u) -1 Z * (v -u) d = exp W (v) - |v -u| 2 + u 2 ∆ = Z * u (v).
The score-function test -which is locally asymptotically uniformly most powerful (LAUMP) in the regular case see [START_REF] Dachian | On hypotheses testing for Poisson processes[END_REF] -does not exist in this non-regular situation. So, we will construct and study the general likelihood ratio test (GLRT), Wald's test (WT) and two Bayesian tests (BT1 and BT2).

General likelihood ratio test

The GLRT is defined by the relations

φ n X (n) = 1 Q(X (n) )>h ε , with Q X (n) = sup ϑ >ϑ 1 L n ϑ , X (n) L n ϑ 1 , X (n) = max L n ϑ n +, X (n) L n ϑ 1 , X (n) , L n ϑ n -, X (n) L n ϑ 1 , X (n) ,
where ϑ n is the maximum likelihood estimator of θ .

To choose the threshold h ε such that φ n X (n) ∈ K ε we need to solve the following equation (under hypothesis H 1 )

P (n) ϑ 1 Q X (n) > h ε = P (n) ϑ 1 sup v>0 Z * n,ϑ 1 (v) > h ε → P sup v>0 Z * (v) > h ε = ε.
For this, we note that the random variable sup v>0 ln Z * (v) has the exponential distribution with parameter 1 see, for example, [START_REF] Baskakova | Tables of distribution of functionals of Brownian motion[END_REF] . This allows us to calculate explicitly the threshold h ε of the GLRT as solution of the equation 1e -ln h ε = 1 -ε, that is, h ε = 1/ε. The power function of the GLRT has the following limit:

β φ n , u = P (n) ϑ u sup v>0 Z * n,ϑ 1 (v) > h ε → P sup v>0 Z * u (v) > h ε .
This limiting power function is obtained below with the help of numerical simulations.

Wald's test To define the WT, let us note that the maximum likelihood estimator ϑ n con- verges in distribution:

(ϕ * n ) -1 ϑ n -ϑ 1 ⇒ ξ * + ,
where the random variable ξ * + is solution of the equation

Z * (ξ * + ) = sup v>0 Z * (v).
Therefore, if we put

φ • n X (n) = 1 (ϕ * n ) -1 ϑ n -ϑ 1 >m ε ,
where m ε is defined by the equation

P ξ * + > m ε = ε, then φ • n ∈ K ε .
We recall the result of [START_REF] Shepp | The joint density of the maximum and its location for a Wiener process with drift[END_REF], that the joint distribution of ln Z * ξ * + , ξ * + has the density

f (y,t) = y √ 2πt 3 exp - y + t 2 2
2t , which allows us to calculate the marginal density of ξ * + as follows:

f (t) = +∞ 0 f (y,t) dy = +∞ 0 y √ t √ 2πt exp - 1 2 y √ t + √ t 2 2 d y √ t = +∞ 0 z √ 2πt exp - 1 2 z + √ t 2 2 dz = +∞ √ t 2 x - √ t 2 √ 2πt exp - x 2 2 dx = - +∞ √ t 2 1 √ 2πt d exp - x 2 2 - +∞ √ t 2 √ t 2 √ 2πt exp - x 2 2 dx = 1 √ 2πt exp - t 8 - 1 2 Φ - √ t 2 ,
where Φ is the distribution function of the standard Gaussian low N (0, 1). So, the threshold m ε can be obtained as the solution of the equation

+∞ m ε 1 √ 2πt exp - t 8 - 1 2 Φ - √ t 2 dt = ε. (9)
The power function of the WT has the following limit:

β φ • n , u = P (n) ϑ u (ϕ * n ) -1 ϑ n -ϑ u + u > m ε → P {ξ * u > m ε -u} ,
where the random variable ξ * u is solution of the equation

Z (ξ * u ) = sup v>-u Z * (v) .
Note that we can also derive another expression of the limiting power function of the WT as follows:

β φ • n , u = P (n) ϑ u (ϕ * n ) -1 ϑ n -ϑ 1 > m ε → P ξ * u,+ > m ε ,
where the random variable ξ * u,+ is solution of the equation

Z ξ * u,+ = sup v>0 Z * u (v) .
The threshold and the limiting power function are obtained below with the help of numerical simulations.

Bayesian tests Suppose now that the parameter ϑ is a random variable with the a priori density p(θ ), ϑ 1 ≤ θ < β . This density is supposed to be continuous and positive. We consider two tests. The first one (BT1) is based on the Bayesian estimator:

φ n X (n) = 1 {(ϕ * n ) -1 ( ϑ n -ϑ 1) >k ε } .
As above, we have the convergence in distribution:

(ϕ * n ) -1 ϑ n -ϑ 1 ⇒ ζ * + ∆ = +∞ 0 v Z * (v) dv +∞ 0 Z * (v) dv
, which allows us to chose the threshold such that φ n ∈ K ε as the solution of the equation

P ζ * + > k ε = ε. ( 10
)
The power function of the BT1 has the following limit:

β φ n , u = P (n) ϑ u (ϕ * n ) -1 ϑ n -ϑ u + u > k ε → P {ζ * u > k ε -u} ,
where the random variable ζ * u is given by

ζ * u = +∞ -u v Z * (v) dv +∞ -u Z * (v) dv
.

Note that we can also derive another expression of the limiting power function of the BT1 as follows:

β φ n , u = P (n) ϑ u (ϕ * n ) -1 ϑ n -ϑ 1 > k ε → P ζ * u,+ > k ε ,
where the random variable ζ * u,+ is given by

ζ * u,+ = +∞ 0 v Z * u (v) dv +∞ 0 Z * u (v) dv
The threshold and the limiting power function are obtained below with the help of numerical simulations.

The second test (BT2) minimizes the mean error. The likelihood ratio is

L X (n) = β ϑ 1 L n θ , X (n) L n ϑ 1 , X (n) p(θ ) dθ = ϕ * n (ϕ * n ) -1 (β -ϑ 1 ) 0 Z * n,ϑ 1 (v) p(ϑ 1 + vϕ * n ) dv.
Hence, we have the following limit:

(ϕ * n ) -1 L X (n) ⇒ p (ϑ 1 ) +∞ 0 exp W (v) - v 2 dv.
Therefore, if we denote

R n = (ϕ * n ) -1 L X (n) p (ϑ 1 )
and chose g ε as solution of the equation

P +∞ 0 exp W (v) - v 2 dv > g ε = ε, the test 1 {R n >g ε } belongs to the class K ε .
Numerical simulations Now, let us carry out some numerical simulations for the GLRT, the WT and the BT1. We take r n = n -0.25 and, in order to simplify the simulations, we take a function ψ n (t) depending neither on n nor on t. More precisely, we consider n independent trajectories

X (n) j = X (n) j (t), t ∈ [0, 4] , j = 1, . . ., n, of an inhomogeneous Poisson process on the interval [0, 4] of intensity function λ (n) ϑ (t) = 1.5 + n -0.25 1 {t>ϑ } , 0 ≤ t ≤ 4,
with ϑ ∈ [2, 4). So, denoting ϑ 1 = 2 and

ϕ * n = ψ(ϑ 1 ) n r 2 n = 1.5 √ n , we have (for v ≥ 0) ln Z * n,ϑ 1 (v) = n ∑ j=1 (ϑ 1 ,ϑ 1 +vϕ * n ] ln 1.5 1.5 + n -0.25 X (n) j (dt) + 1.5 v n 0.25 = ln 1.5 1.5 + n -0.25 n ∑ j=1 X (n) j (ϑ 1 + vϕ * n ) -X (n) j (ϑ 1 ) + 1.5 v n 0.25 .
Some realizations of Z * n,ϑ 1 can be found in Figure 1. Recall that the threshold h ε = 1/ε of the GLRT is known explicitly. We obtain the threshold m ε of the WT by numerically solving the equation ( 9), while the threshold k ε of the BT1 is obtained from the equation (10) by means of numerical simulations of the random variable ζ * + . Some values of the thresholds m ε and k ε are given in Table 1. To illustrate the convergence of power functions of different tests to their limits, we present in Figure 2 the power functions for n = 100 (r n = 0.3162) and n = 300 (r n = 0.2403), as well as the limiting power functions. All these power functions are obtained by means of numerical simulations. Note that the values of u greater than 2(ϕ * n ) -1 correspond to θ u = θ 1 + uϕ * n > 4, which means that there is no longer jump in intensity function on the interval [0, 4]. This explains the fact that for n = 100, the power functions are constant for u > 2(ϕ * 100 ) -1 ≈ 13.33. Comparison of the limiting power functions Let us fix an alternative u 1 > 0 and consider the testing problem with two simple hypotheses

H 1 : u = 0, H u 1 2 : u = u 1 .
Remind that in this situation the most powerful test is the Neyman-Pearson test (N-PT). Of course, it is impossible to use the N-PT in our initial problem, because it depends on the value u 1 under alternative which is unknown. However, its power (considered as function of u 1 ) gives an upper bound (Neyman-Pearson envelope) for the power functions of all the tests. Therefore, it is interesting to compare the power functions of different tests not only one with another, but also with the power of the N-PT.

The N-PT is given by

φ * n X (n) = 1 Z * n,ϑ 1 (u 1 )>d ε + q ε 1 Z * n,ϑ 1 (u 1 )=d ε ,
where d ε and q ε are solution of the equation

P (n) ϑ 1 Z * n,ϑ 1 (u 1 ) > d ε + q ε P (n) ϑ 1 Z * n,ϑ 1 (u 1 ) = d ε = ε. (11)
Recall that the likelihood ratio Z * n,ϑ 1 (u 1 ) under hypothesis H 1 converges to the following limit Hence, in the asymptotic setting, the equation ( 11) can be replaced by the equation

Z * n,ϑ 1 (u 1 ) ⇒ Z * (u 1 ) = exp W (u 1 ) - u 1 2 .
P Z * (u 1 ) > d ε + q ε P Z * (u 1 ) = d ε = ε
and, since Z * (u 1 ) is a continuous random variable, we can put q ε = 0 and find the threshold d ε as the solution of the equation

P Z * (u 1 ) > d ε = ε. Note that P Z * (u 1 ) > d ε = P W (u 1 ) > ln d ε + u 1 2 = P ζ > ln d ε + u 1 2 √ u 1 ,
where ζ ∼ N (0, 1). Therefore, denoting z ε the quantile of order 1 -ε of the standard Gaussian law P (ζ > z ε ) = ε , the threshold d ε is given by

d ε = e z ε √ u 1 - u 1 2 .
Under alternative H u 1 2 , we have

Z * n,ϑ 1 (u 1 ) = L n ϑ 1 + u 1 ϕ * n , X (n) L n ϑ 1 , X (n) = L n ϑ 1 + u 1 ϕ * n -u 1 ϕ * n , X (n) L n ϑ 1 + u 1 ϕ * n , X (n) -1 ⇒ Z * (-u 1 ) -1 ,
which allows us to obtain the limiting power of the N-PT as follows:

β φ * n = P (n) ϑ 1 +u 1 ϕ * n Z * n,ϑ 1 (u 1 ) > d ε → P Z * (-u 1 ) -1 > d ε = P exp -W (-u 1 ) + u 1 2 > d ε = P W (u 1 ) > ln d ε - u 1 2 = P ζ > ln d ε -u 1 2 √ u 1 = P ζ > z ε - √ u 1 .
So, the limiting Neyman-Pearson envelope is given by

β (u) = P ζ > z ε - √ u = 1 -Φ z ε - √ u ,
where, as before, Φ is the distribution function of the standard Gaussian low.

The limiting power functions of the GLRT, of the WT and of the BT1 are obtained by means of numerical simulations and are presented in Figure 3 together with the limiting Neyman-Pearson envelope β (u). We can observe that the limiting power function of the GLRT is the closest to the limiting Neyman-Pearson envelope for small values of u, while the limiting power function of the BT1 is the one that tends to 1 (as u becomes large) the most quickly. We can also see that for ε = 0.05 the limiting power functions of the WT and of the BT1 are close (especially when u is small). Finally, we need to say that all these limiting power functions are perceptibly below the limiting Neyman-Pearson envelope, and that the choice of the asymptotically optimal test remains an open question.

Proofs of the lemmas

The proofs of Lemmas 2-4 in the case r = 0, as well as the proof of Lemma 5, are similar to the fixed jump size case and hence are omitted the interested reader can see, for example, [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF][START_REF] Kutoyants | Statistical Inference for Spatial Poisson Processes[END_REF] .

Proof of Lemma 2 in the case r = 0 First we study the convergence of 2-dimensional distributions. For this, consider the distribution of the vector Z n,ϑ (u 1 ), Z n,ϑ (u 2 ) with some fixed u 1 , u 2 ∈ R. The characteristic function of the natural logarithm of this vector can be written as follows see, for example, [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF] :

E (n) ϑ exp it 1 ln Z n,ϑ (u 1 ) + it 2 ln Z n,ϑ (u 2 ) = exp n τ 0 exp it 1 ln λ (n) ϑ +u 1 ϕ n (t) λ (n) ϑ (t) + it 2 ln λ (n) ϑ +u 2 ϕ n (t) λ (n) ϑ (t) -1 -it 1 λ (n) ϑ +u 1 ϕ n (t) λ (n) ϑ (t) -1 -it 2 λ (n) ϑ +u 2 ϕ n (t) λ (n) ϑ (t) -1 λ (n) ϑ (t) dt = exp A n,ϑ (u 1 , u 2 ,t)
with an evident notation. We will consider the case u 2 > u 1 ≥ 0 only (the other cases can be treated in a similar way). In this case, we have

A n,ϑ (u 1 , u 2 ,t) = n ϑ +u 1 ϕ n ϑ exp (it 1 + it 2 ) ln ψ n (t) ψ n (t) + r n - 1
-(it 1 + it 2 ) ψ n (t) ψ n (t) + r n -1 ψ n (t) + r n dt + n ϑ +u 2 ϕ n ϑ +u 1 ϕ n exp it 2 ln ψ n (t) ψ n (t) + r n -1 -it 2 ψ n (t) ψ n (t) + r n -1 ψ n (t) + r n dt = nI 1 + nI 2
with evident notations. Using the mean value theorem for the integrals I 1 and I 2 , it is possible to find some

s n ∈ (ϑ , ϑ + u 1 ϕ n ) and v n ∈ (ϑ + u 1 ϕ n , ϑ + u 2 ϕ n ) such that nI 1 = u 1 r 2 n exp{i(t 1 + t 2 ) ln g n (s n )} -1 -i(t 1 + t 2 ) g n (s n ) -1 ψ n (s n ) + r n and nI 2 = u 2 -u 1 r 2 n exp{it 2 ln g n (v n )} -1 -it 2 g n (v n ) -1 ψ n (v n ) + r n ,
where we have denoted g n (t) = ψ n (t) ψ n (t)+r n = 1 -r n ψ n (t)+r n .

As s n → ϑ , using the condition C3 we obtain lim n→+∞ ψ n (s n ) = ψ(ϑ ). So,

nI 1 ∼ u 1 ψ(ϑ ) r 2 n exp{i(t 1 + t 2 ) lng n (s n )} -1 -i(t 1 + t 2 ) g n (s n ) -1 .
As r n → 0 and ℓ ≤ ψ n (t) + r n ≤ L, we have g n (s n ) -1 = O(r n ) → 0. So, using Taylor expansion we get

ln g n (s n ) = ln 1 + g n (s n ) -1 = g n (s n ) -1 - 1 2 g n (s n ) -1 2 + o r 2 n (ψ n (s n ) + r n ) 2 = g n (s n ) -1 - 1 2 g n (s n ) -1 2 + o(r 2 n ).
In particular, ln

g n (s n ) = O(r n ) and ln g n (s n ) 2 = g n (s n ) -1 2 + o(r 2 n ).
Using Taylor expansion once more, we obtain

exp it ln g n (s n ) = 1 + it ln g n (s n ) - t 2 2 ln g n (s n ) 2 + o(r 2 n ).
So,

nI 1 ∼ u 1 ψ(ϑ ) r 2 n -i(t 1 + t 2 ) g n (s n ) -1 2 2 - (t 1 + t 2 ) 2 2 g n (s n ) -1 2 + o(r 2 n ) = u 1 ψ(ϑ ) r 2 n - i(t 1 + t 2 )r 2 n 2(ψ(s n ) + r n ) 2 - (t 1 + t 2 ) 2 r 2 n 2(ψ(s n ) + r n ) 2 + o(r 2 n ) → u 1 ψ(ϑ ) - i(t 1 + t 2 ) 2 - (t 1 + t 2 ) 2 2 .
Similarly, we can show that

nI 2 → u 2 -u 1 ψ(ϑ ) - it 2 2 - t 2 2 2 ,
and hence

E (n) ϑ exp it 1 ln Z n,ϑ (u 1 ) + it 2 ln Z n,ϑ (u 2 ) → exp - u 2 -u 1 2ψ(ϑ ) (it 2 + t 2 2 ) - u 1 2ψ(ϑ ) i(t 1 + t 2 ) + (t 1 + t 2 ) 2 . (12)
For all u > 0, we introduce the σ -algebra

F u = σ W (v), 0 ≤ v ≤ u and write E exp it 1 ln Z ϑ (u 1 ) + it 2 ln Z ϑ (u 2 ) = E exp i(t 1 + t 2 ) ln Z ϑ (u 1 ) E exp it 2 ln Z ϑ (u 2 ) -ln Z ϑ (u 1 ) F u 1 = exp - (t 1 + t 2 ) 2 2ψ(ϑ ) u 1 - i(t 1 + t 2 ) 2ψ(ϑ ) u 1 - t 2 2 2ψ(ϑ ) (u 2 -u 1 ) - it 2 2ψ(ϑ ) (u 2 -u 1 ) .
Combining this with [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], we obtain the convergence of 2-dimensional distributions. The convergence of three and more dimensional distributions can be carried out in a similar way, and the uniformity with respect to ϑ is obvious.

⊓ ⊔

Proof of Lemma 3 in the case r = 0 We will consider the case u 2 ≥ u 1 ≥ 0 only (the other cases can be treated in a similar way). According to [14, Lemma 1.1.5], we have

E (n) ϑ Z 1/2 n,ϑ (u 1 ) -Z 1/2 n,ϑ (u 2 ) 2 ≤ n τ 0 λ (n) ϑ +u 1 ϕ n (t) -λ (n) ϑ +u 2 ϕ n (t) 2 dt = n ϑ +u 2 ϕ n ϑ +u 1 ϕ n ψ n (t) + r n -ψ n (t) 2 dt = n ϑ +u 2 ϕ n ϑ +u 1 ϕ n r 2 n ψ n (t) + r n + ψ n (t) 2 dt. As λ (n)
ϑ is uniformly separated from zero, we have

ψ n (t) + r n + ψ n (t) 2 ≥ √ ℓ + √ ℓ 2 = 4ℓ,
and hence

E (n) ϑ Z 1/2 n,ϑ (u 1 ) -Z 1/2 n,ϑ (u 2 ) 2 ≤ n ϑ +u 2 ϕ n ϑ +u 1 ϕ n r 2 n 4ℓ dt = 1 4ℓ |u 1 -u 2 | .
So, the required inequality holds with C = 1 4ℓ .

⊓ ⊔

Proof of Lemma 4 in the case r = 0 We will consider the case u ≥ 0 only (the other case can be treated in a similar way). According to [14, Lemma 1.1.5], we have

E (n) ϑ Z 1/2 n,ϑ (u) = exp - n 2 τ 0 λ (n) ϑ +uϕ n (t) -λ (n) ϑ (t) 2 dt = exp - n 2 ϑ +uϕ n ϑ ψ n (t) -ψ n (t) + r n 2 dt = exp - n 2 ϑ +uϕ n ϑ r 2 n ψ n (t) + ψ n (t) + r n 2 dt . As λ (n)
ϑ is uniformly bounded, we have

ψ n (t) + r n + ψ n (t) 2 ≤ √ L + √ L 2 = 4L,
and hence

E (n) ϑ Z 1/2 n,ϑ (u) ≤ exp - n 2 ϑ +ϕ n u ϑ r 2 n 4L dt = exp - 1 8L |u| .
So, the required inequality holds with k * = 1 8L .

⊓ ⊔

Proof of Lemma 6 Using Markov inequality, we get

P (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) > ε ≤ 1 ε 2 E (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) 2 .
First we consider the case u 1 , u 2 ≥ 0 (and say u 2 ≥ u 1 ). In this case, we have

ln Z n,ϑ (u 2 ) -ln Z n,ϑ (u 1 ) = n ∑ j=1 ϑ +u 2 ϕ n ϑ +u 1 ϕ n ln ψ n (t) ψ n (t) + r n dX (n) j (t) + n ϑ +u 2 ϕ n ϑ +u 1 ϕ n r n dt = n ∑ j=1 ϑ +u 2 ϕ n ϑ +u 1 ϕ n ln ψ n (t) ψ n (t) + r n dY (n) j (t) + n ϑ +u 2 ϕ n ϑ +u 1 ϕ n ψ n (t) + r n ln ψ n (t) ψ n (t) + r n + r n dt, where Y (n) j
is the centered version of the process X

(n) j . Since the stochastic integrals with respect to Y (n) j , j = 1, . . . , n, are independent and has mean zero, we obtain In the case u 1 , u 2 ≤ 0, proceeding similarly, we obtain the same inequality. Finally, in the case u 1 u 2 < 0 (say u 1 < 0 and u 2 > 0), we obtain Note that this final inequality holds for all the three cases, and so In order to estimate the first term, first let us note that the Markov process Z n,ϑ (u), u ≥ 0, is a martingale. Indeed, for any v ≥ u ≥ 0, using the representation [START_REF] Dachian | On limiting likelihood ratio processes of some change-point type statistical models[END_REF] ⊓ ⊔

E (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) 2 = n E (n) ϑ ϑ +u 2 ϕ n ϑ +u 1 ϕ n ln ψ n (t) ψ n (t) + r n dY (n) j (t) 2 + n 2 ϑ +u 2 ϕ n ϑ +u 1 ϕ n ψ n (t) + r n ln ψ n (t) ψ n (t) + r n + r n dt 2 = E 1 +
E (n) ϑ ln Z n,
P (n) ϑ ln Z n,ϑ (u 1 ) -lnZ n,ϑ (u 2 ) > ε ≤ 25 |u 1 -u 2 | 8ε 2 ℓ + (u 2 -u 1 ) 2 ε 2

ϑ

  (t) can be written as λ (n) ϑ (t) = ψ n (t) + r n 1 {t>ϑ } , where the function ψ n is continuous on [0, τ].

  du . Let us note that ξ ϑ d = ψ(ϑ ) ξ * and ζ ϑ d = ψ(ϑ ) ζ * . Now we can state the following theorem giving an asymptotic lower bound on the risk of all the estimators of ϑ .

Theorems 3 -

 3 5 follow from the properties of the normalized likelihood ratio established in Section 3. More precisely, Theorem 5 is a consequence of Lemmas 2-4 and [11, Theorem 1.10.2]. Having the properties of the Bayesian estimators given in Theorem 5, we can cite [11, Theorem 1.9.1] to provide the proof of Theorem 3. Finally, the proof of Theorem 4 can be carried out following the standard argument of[START_REF] Ibragimov | Statistical Estimation. Asymptotic Theory[END_REF] Chapters 5.3 and 5.4] which is based on the weak convergence established in Theorem 1 together with the inequality[START_REF] Fujii | On weak convergence of the likelihood ratio process in multi-phase regression models[END_REF].

Fig. 1

 1 Fig. 1 Some realization of Z * n,ϑ 1 (v)

Fig. 2

 2 Fig. 2 Power functions of GLRT, WT and BT1

Fig. 3

 3 Fig. 3 Comparison of limiting power functions for ε = 0.05 and ε = 0.4

ℓ 2 for all u 1 ,8ε 2 ℓ + h 2 ε 2 ℓ 2

 12 u 2 ∈ U n and sufficiently large values of n. Hence,lim n→+∞ sup |u 1 -u 2 |<h P (n) ϑ ln Z n,ϑ (u 1 ) -ln Z n,ϑ (u 2 ) > ε ≤ 25h → 0as h → 0, and so, the lemma is proved. ϑ (u) > e -bD ≤ P (n) ϑ sup u>D Z n,ϑ (u) > e -bD + P ϑ (u) > e -bD .

  ϑ (u) > e -bD = P (n) ϑ sup t>0 X(t) > e -bD/2 ≤ e bD/2 EX(0) = e bD/2 EZ 1/2 n,ϑ (D).So, using Lemma 4, we can majorate the first term asP (n) ϑ sup u>D Z n,ϑ (u) > e -bD ≤ e bD/2 e -k * D ≤ e -bD ,where the last inequality is valid if b ≤ 2k * /3.For the second term, in a similar manner (and under the same condition b ≤ 2k * /3) we obtain the bound P ϑ (u) > e -bD ≤ e -bD , and so, the required inequality holds with C = 2 and any b ∈ (0, 2k * /3].

Table 1

 1 Thresholds of WT and BT1

	ε	0.001	0.005	0.01	0.05	0.1	0.2
	m ε	30.336	20.686	14.886	7.282	4.531	2.236
	k ε	24.877	17.588	16.782	8.582	5.573	3.024

  E 2 with obvious notations. Using elementary inequalities ln(1 + x) ≤ x and ln(1 + x) ≥ xx 2 /2 for |x| < 1/2, for sufficiently large values of n (such that r n ψ n (t)+r n < r n ℓ < 1/2) we obtain ln Z n,ϑ (u 1 ) -lnZ n,ϑ (u 2 )

	and hence
			E 2 = n 2		ϑ +u 2 ϕ n ϑ +u 1 ϕ n	ψ n (t) + r n ln	ψ n (t) + r n ψ n (t)	+ r n dt	2
				≤ n 2		ϑ +u 2 ϕ n ϑ +u 1 ϕ n	r 2 n 2ℓ	dt	2	=	(u 2 -u 1 ) 2 4ℓ 2	.
	Thus, for sufficiently large values of n, we have
	E	(n)								2 ≤	25 |u 1 -u 2 | 16ℓ	+	(u 2 -u 1 ) 2 4ℓ 2	.
	-	r n ψ n (t) + r n	-	r 2 n 2 ψ n (t) + r n	2 ≤ ln	ψ n (t) ψ n (t) + r n	≤ -	r n ψ n (t) + r n	.
					ϑ +u 2 ϕ n ϑ +u 1 ϕ n		ln	ψ n (t) ψ n (t) + r n	2	(ψ n (t) + r n ) dt
			≤ n	ϑ +u 2 ϕ n ϑ +u 1 ϕ n	r 2 n ψ n (t) + r n	dt ≤ n	(u 2 -u 1 )r 2 n ϕ n ℓ	=	|u 1 -u 2 | ℓ	.
	As to the case r n ≥ 0, as	r n ψ n (t)+r n < 1/2, we have
	E 1 = n	ϑ +u 2 ϕ n ϑ +u 1 ϕ n	ln	ψ n (t) ψ n (t) + r n	2	(ψ n (t) + r n ) dt
	≤ n	ϑ +u 2 ϕ n ϑ +u 1 ϕ n	r 2 n ψ n (t) + r n	+	r 3 n (ψ n (t) + r n ) 2 +	r 4 n 4(ψ n (t) + r n ) 3 dt
	≤ n	ϑ +u 2 ϕ n ϑ +u 1 ϕ n	r 2 n ψ n (t) + r n	1 +	1 2	+	1 16	dt ≤	25 |u 1 -u 2 | 16ℓ	.
	For E 2 , we have
	-	r 2 n 2ℓ	≤ -	r 2 n 2(ψ n (t) + r n )	≤ ψ n (t) + r n ln	ψ n (t) ψ n (t) + r n	+ r n ≤ 0,

For E 1 , if r n ≤ 0, we obtain

E 1 = n ϑ

  ϑ (u 1 )ln Z n,ϑ (u 2 ) 2 ≤ 2 E

			(n) ϑ ln Z n,ϑ (u 1 )	2 + 2 E	(n) ϑ ln Z n,ϑ (u 2 )	2
	≤	25 |u 1 | 8ℓ	+	u 2 1 2ℓ 2 +	25 |u 2 | 8ℓ	+	u 2 2 2ℓ 2
	=	25 8ℓ	|u 1 | + |u 2 | +	1 2ℓ 2 u 2 1 + u 2 2
	≤	25 |u 1 -u 2 | 8ℓ	+	(u 2 -u 1 ) 2 ℓ 2	.

  we can write E Z n,ϑ (v) Z n,ϑ (u) (t + D), t ≥ 0, is a supermartingale and, using the maximal inequality for positive supermartingales see, for example, Revuz and Yor [16, Exercise 2.1.15] , we get

	1/2 n,ϑ P Hence, the process X(t) = Z (n) ϑ
	= E exp	n ∑ j=1 (ϑ +uϕ n ,ϑ +vϕ n ]	ln	ψ n (t) ψ n (t) + r n	X	(n) j (dt) +	v -u r n	Z n,ϑ (u) Z n,ϑ (u)
	= Z n,ϑ (u) exp	v -u r n	n ∏ j=1	E exp	(ϑ +uϕ n ,ϑ +vϕ n ]	ln	ψ n (t) ψ n (t) + r n	X	(n) j (dt)
	= Z n,ϑ (u) exp	v -u r n	exp n	ϑ +vϕ n ϑ +uϕ n	ψ n (t) ψ n (t) + r n	-1 (ψ n (t) + r n ) dt
	= Z n,ϑ (u) exp	v -u r n	exp -n	ϑ +vϕ n ϑ +uϕ n	r n dt
	= Z n,ϑ (u) exp	v -u r n	exp -n(v -u)ϕ n r n = Z n,ϑ (u).

Acknowledgements This study was partially supported by the Russian Science Foundation (research project no. 14-49-00079). The authors would like to thank the anonymous reviewers for their helpful and constructive comments, which greatly contributed to the improvement of the paper.