On Smooth Change-Point Location Estimation for Poisson Processes
OOn Smooth Change-Point Location Estimation forPoisson Processes
A. Amiri and S. Dachian University of Lille, CNRS, UMR 8524 — Laboratoire Paul Painlev´e, F–59000 Lille, France Tomsk State University, International Laboratory of Statistics of Stochastic Processes andQuantitative Finance, 634050 Tomsk, Russia
Abstract
We are interested in estimating the location of what we call “smooth change-point” from n independent observations of an inhomogeneous Poisson process. Thesmooth change-point is a transition of the intensity function of the process from onelevel to another which happens smoothly, but over such a small interval, that itslength δ n is considered to be decreasing to 0 as n → + ∞ . We show that if δ n goesto zero slower than 1 /n , our model is locally asymptotically normal (with a ratherunusual rate (cid:112) δ n /n ), and the maximum likelihood and Bayesian estimators areconsistent, asymptotically normal and asymptotically efficient. If, on the contrary, δ n goes to zero faster than 1 /n , our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimatorsare consistent, converge at rate 1 /n , have non-Gaussian limit distributions and areasymptotically efficient. All these results are obtained using the likelihood ratioanalysis method of Ibragimov and Khasminskii, which equally yields the convergenceof polynomial moments of the considered estimators. However, in order to study themaximum likelihood estimator in the case where δ n goes to zero faster than 1 /n , thismethod cannot be applied using the usual topologies of convergence in functionalspaces. So, this study should go through the use of an alternative topology and willbe considered in a future work. Keywords: inhomogeneous Poisson process, smooth change-point, maximum like-lihood estimator, Bayesian estimators, local asymptotic normality, asymptotic effi-ciency
AMS subject classification:
This paper lies within the realm of statistical inference for inhomogeneous Poisson pro-cesses. Recall that X = (cid:0) X ( t ) , ≤ t ≤ T (cid:1) is an inhomogeneous Poisson process (on an1 a r X i v : . [ m a t h . S T ] S e p nterval [0 , T ]) of intensity function λ ( t ), 0 ≤ t ≤ T , if X (0) = 0 and the increments of X on disjoint intervals are independent Poisson random variables: P (cid:8) X ( t ) − X ( s ) = k (cid:9) = (cid:0)(cid:82) ts λ ( t ) d t (cid:1) k k ! exp (cid:26) − (cid:90) ts λ ( t ) d t (cid:27) . The model of inhomogeneous Poisson process is at the same time simple enough to al-low the use of the likelihood ratio analysis, and sufficiently reach to modelize variousrandom phenomena in diverse applied fields, such as biology, communication, seismology,astronomy, reliability theory, and so on (see, for example, Cox and Lewis (1966), Thomp-son (1988), Snyder and Miller (1991), Streit (2010), Sarkar (2016), as well as Cha andFinkelstein (2018)).We are interested in the problem of estimation of the location θ , where the (elsewheresmooth) intensity function of an inhomogeneous Poisson process switches from one level(say λ ) to another (say λ + r ). This transition can happen in several ways. The intensityfunction can switch from λ to λ + r instantaneously (change-point case), as for examplein λ θ ( t ) = λ + r { t ≥ θ } . It can also go from λ to λ + r smoothly over a small interval of some fixed length δ > λ θ ( t ) = λ + rδ ( t − θ ) { θ ≤ t<θ + δ } ( t ) + r { t ≥ θ + δ } ( t ) . As an intermediate case, we can mention the case of the cusp type singularity, where theintensity function goes from λ to λ + r continuously over a small interval of some fixedlength δ but has an infinite derivative at some point of this interval, as for example in λ θ ( t ) = λ + rδ κ ( t − θ ) κ { θ ≤ t<θ + δ } ( t ) + r { t ≥ θ + δ } ( t ) , where κ ∈ (0 , /
2) is the order of the cusp. The three above intensity functions areillustrated in Figure 1.In all these cases, the estimation problem is considered in some asymptotic setting,such as n → + ∞ independent observations on a fixed interval, large observation intervalasymptotics (an observation on the interval [0 , nτ ] with τ -periodic intensity function), largeintensity asymptotics (an observation on a fixed interval with intensity function multipliedby n ), and so on.In the smooth case, the statistical model is regular. The regular statistical models forPoissonian observations were studied by Kutoyants in (1979, 1984, 1998). It was shownthat such models are locally asymptotically normal, and that the maximum likelihood andBayesian estimators (for any continuous strictly positive prior density q ) are consistent,asymptotically normal (with classic rate 1 / √ n ) and asymptotically efficient.In the change-point case, studied by Kutoyants in (1984, 1998), the properties of theestimators are essentially different. The maximum likelihood and Bayesian estimators2 q q + dl l + r change−pointsmoothcusp Figure 1: change-point, smooth and cusp (with κ = 1 /
4) casesare consistent, converge at a faster rate 1 /n , their limit distributions are given by some(different) functionals of a two-sided Poisson process, and only the Bayesian estimators areasymptotically efficient.Finally, the cusp case was studied by Dachian in (2003). In this case, the maximumlikelihood and Bayesian estimators are consistent, converge at rate 1 /n (2 κ +1) (which is fasterthan 1 / √ n and slower than 1 /n ), their limit distributions are given by some (different)functionals of a two-sided fractional Brownian motion, and only the Bayesian estimatorsare asymptotically efficient.Let us note here that all the above cited studies were carried out using the likelihoodratio analysis method introduced by Ibragimov and Khasminskii in (1981), which equallyyields the convergence of polynomial moments of the considered estimators.Note also that recently the problem of source localization on the plane by observationsof Poissonian signals from several detectors was considered in all the three cases (smooth,change-point and cusp) in Chernoyarov and Kutoyants (2020), Farinetto, Kutoyants andTop (2020) and Chernoyarov, Dachian and Kutoyants (2020), respectively.In this paper, we consider the situation, which we call smooth change-point , where theintensity function goes from λ to λ + r smoothly, but over such a small interval, that itslength is considered to be decreasing to 0 as n → + ∞ . Such an intensity function can, forexample, be given by λ ( n ) θ ( t ) = λ + rδ n ( t − θ ) { θ ≤ t<θ + δ n } ( t ) + r { t ≥ θ + δ n } ( t ) , where δ n →
0. Note that the intensity function now depends on n , and so, we are in ascheme of series (triangular array) framework.The main result of the paper is that there is a “phase transition” in the asymptoticbehavior of the estimators depending on the rate at which δ n →
0. More precisely, we show3hat if δ n goes to zero slower than the “critical” rate 1 /n , the behavior resembles that ofthe smooth case, and if δ n goes to zero faster than 1 /n , the behavior is exactly the same asin the change-point case. We call these two situations slow case and fast case , respectively.More specifically, in the slow case we show that our model is locally asymptoticallynormal, and that the maximum likelihood and Bayesian estimators are consistent, asymp-totically normal and asymptotically efficient. It should be noted here that all these asymp-totic results use a rather unusual rate (cid:112) δ n /n , which is faster than the rate 1 / √ n of thesmooth case and slower than the rate 1 /n of the change-point case.As to the fast case, we show that the asymptotic behavior of the Bayesian estimators isexactly the same as in the change-point model: they are consistent, converge at rate 1 /n ,their limit distribution is given by a functional of a two-sided Poisson process, and they areasymptotically efficient. In our opinion, these results justify the (successful) use of change-point models for real applications, despite the fact that physical systems can not switchimmediately (discontinuously) from one level to another. Indeed, if the transition happensquickly enough, it seems more appropriate to use the fast case of our smooth change-pointmodel, and yet it yields the same asymptotic behavior (at least for the Bayesian estimators,although we conjecture that it is also true for the maximum likelihood estimator).Let us note that all our results were obtained using the likelihood ratio analysis methodof Ibragimov and Khasminskii, which equally yields the convergence of polynomial momentsof the considered estimators. On the other hand, for the study of the maximum likelihoodestimator, this method needs the convergence of the normalized likelihood ratio in somefunctional space, and up to the best of our knowledge, until now it was only applied usingeither the space C ( R ) of continuous functions on R vanishing at ±∞ equipped with thetopology induced by the usual sup norm, or the Skorokhod space D ( R ) of c`adl`ag functionsvanishing at ±∞ equipped with the usual Skorokhod topology. However, we will see thatin the fast case this convergence can not take place in neither of these topologies, as bothof them do not allow the convergence of continuous functions to a discontinuous limit. So,the study of the maximum likelihood estimator in the fast case should go through the useof an alternative topology and will be considered in a future work.Another possible perspective is to study the behavior of the estimators in the criticalcase δ n = c/n . Also, for models where the transition of the intensity function from λ to λ + r is continuous over an interval of length δ but has a cusp of order κ at some point ofthis interval, it can be interesting to study the situations (somewhat similar to our model)where δ = δ n → κ = κ n → We consider the model of observation of n ∈ N ∗ independent realizations of an inhomoge-neous Poisson process. Let 0 < α < β < τ be some known constants, and ψ be some knownstrictly positive continuous function on [0 , τ ]. Let also r > − min ≤ t ≤ τ ψ ( t ) be some knownconstant, ( δ n ) n ∈ N be some known sequence decreasing to 0, and θ ∈ Θ = ( α, β ) be a one-dimensional unknown parameter that we want to estimate in the asymptotics n → + ∞ .4e observe X ( n ) = ( X , . . . , X n ), where X j = (cid:0) X j ( t ) , ≤ t ≤ τ (cid:1) , j = 1 , . . . , n , are inde-pendent Poisson processes on the interval [0 , τ ] with intensity function λ θ = λ ( n ) θ , θ ∈ Θ,given by λ ( n ) θ ( t ) = ψ ( t ) + rδ n ( t − θ ) [ θ,θ + δ n [ ( t ) + r [ θ + δ n ,τ ] ( t ) , ≤ t ≤ τ. (1)This function, in an important particular case ψ ≡ λ >
0, is presented in Figure 2. tq q + d n l l + r Figure 2: intensity function λ ( n ) θ with ψ ≡ λ Note that our model of observation is equivalent to observing a single realizationon [0 , nτ ] of an inhomogeneous Poisson process X = (cid:0) X ( t ) , ≤ t ≤ nτ (cid:1) of τ -periodicintensity function equal to λ ( n ) θ on the first period (large observation interval asymptotics).Also, it is equivalent to observing a single realization on [0 , τ ] of an inhomogeneous Pois-son process Y ( n ) = (cid:0) Y ( n ) ( t ) , ≤ t ≤ τ (cid:1) of intensity function Λ ( n ) θ = n λ ( n ) θ (large intensityasymptotics).Recall that regular models of Poissonian observations were treated previously and shownto be locally asymptotically normal (LAN) by Kutoyants in (1979, 1984, 1998) (see alsoDachian, Kutoyants and Yang (2016a), where the corresponding hypothesis testing problemwas considered). An example of such a regular model is the model with intensity function λ θ given as λ ( n ) θ in (1), but with δ n replaced by a strictly positive constant δ : λ θ ( t ) = ψ ( t ) + rδ ( t − θ ) [ θ,θ + δ [ ( t ) + r [ θ + δ,τ ] ( t ) , ≤ t ≤ τ. (2)In this case, the intensity function is continuous and do not depend on n .Recall also that various singular models of Poisonnian observations were already treatedpreviously. The change-point case was studied by Kutoyants in (1984, 1998), the cusp case5as considered by Dachian in (2003), and the cases of 0-type and ∞ -type singularities wereinvestigated by Dachian in (2011) (see also Dachian, Kutoyants and Yang (2016b), wherethe hypothesis testing problem was considered for different singular cases). An exampleof a change-point model is the model with intensity function λ θ given as λ ( n ) θ in (1), butwith δ n replaced by 0: λ θ ( t ) = ψ ( t ) + r { t ≥ θ } , ≤ t ≤ τ. (3)In this case, the intensity function is discontinuous and do not depend on n .For our model, the intensity function λ ( n ) θ is continuous for all n ∈ N ∗ , but its limitas n → + ∞ is discontinuous. It is, in some sense, the inverse of the case treated in Dachianand Yang (2015), where the intensity function was supposed to have a discontinuity thatdisappears as n → + ∞ .We denote P ( n ) θ the probability measure corresponding to X ( n ) . The likelihood, withrespect to the measure P ∗ corresponding to n independent homogeneous Poisson processesof unit intensity, is given (see, for example, Liptser and Shiryaev (2001)) by L (cid:0) θ, X ( n ) (cid:1) = d P θ (cid:0) X ( n ) (cid:1) d P ∗ = exp (cid:40) n (cid:88) j =1 (cid:90) τ ln (cid:0) λ θ ( t ) (cid:1) d X j ( t ) − n (cid:90) τ (cid:0) λ θ ( t ) − (cid:1) d t (cid:41) , θ ∈ Θ . As estimators of the unknown parameter θ , we consider the maximum likelihood esti-mator (MLE) and the Bayesian estimators (BEs). The MLE ˆ θ n is given byˆ θ n = argsup θ ∈ Θ L (cid:0) θ, X ( n ) (cid:1) , and the BE ˜ θ n for quadratic loss and prior density q is given by˜ θ n = (cid:82) βα θ q ( θ ) L (cid:0) θ, X ( n ) (cid:1) d θ (cid:82) βα q ( θ ) L (cid:0) θ, X ( n ) (cid:1) d θ . Both in the regular and singular cases cited above, the study of the asymptotic behaviorof the MLE and of the BEs was carried out using the likelihood ratio analysis methodintroduced by Ibragimov and Khasminskii in (1981). This method consist in first studyingthe normalized likelihood ratio given by Z n ( u ) = Z ( θ ) n ( u ) = d P θ + uϕ n (cid:0) X ( n ) (cid:1) d P θ = L (cid:0) θ + uϕ n , X ( n ) (cid:1) L (cid:0) θ, X ( n ) (cid:1) = exp (cid:40) n (cid:88) j =1 (cid:90) τ ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) − n (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ ( t ) (cid:1) d t (cid:41) , u ∈ U n , ϕ n is some sequence decreasing to 0, called likelihood normalization rate , and U n = ] ϕ − n ( α − θ ) , ϕ − n ( β − θ )[. This rate must be chosen so that the process Z n converges(in some sense) to a non-degenerate (not identically equal to 1) limit process defined onthe whole real line (note that U n ↑ R ), called limit likelihood ratio . Then, the properties ofthe MLE and of the BEs are deduced.In the regular case (see Kutoyants (1979, 1984, 1998)), the likelihood normalizationrate can be chosen as ϕ n = 1 √ n . Note that in this case the processes Z n , n ∈ N , can be extended to the whole real lineso that their trajectories almost surely belong to the space C ( R ) of continuous functionson R vanishing at ±∞ . The process Z n converge, in C ( R ) equipped with the usual supnorm, to the process Z ◦ I ( θ ) , where I ( θ ) = (cid:90) τ (cid:0) ˙ λ θ ( t ) (cid:1) λ θ ( t ) d t (here ˙ λ θ ( t ) denotes the derivative of λ θ ( t ) w.r.t. θ ) is the Fisher information, and for any F ∈ R , the process Z ◦ F is defined by Z ◦ F ( u ) = exp (cid:26) u ∆ F − u F (cid:27) , u ∈ R . (4)Here and in the sequel ∆ F ∼ N (0 , F ). In fact, the model is LAN (with classic rate 1 / √ n )in this case. Of course, we could have also chosen the rate ϕ n = 1 / (cid:112) nI ( θ ), in which casethe limit likelihood ratio process would be Z ◦ ( u ) = exp (cid:26) u ∆ − u (cid:27) , u ∈ R . Then, the MLE and the BEs (for any continuous strictly positive prior density q ) areconsistent, are asymptotically normal with rate 1 / √ n : √ n (cid:0) ˆ θ n − θ (cid:1) = ⇒ ∆ I ( θ ) and √ n (cid:0) ˜ θ n − θ (cid:1) = ⇒ ∆ I ( θ ) , we have the convergence of polynomial moments, and both the estimators are asymptoti-cally efficient.Note also that in the case of the intensity function given by (2), using the change ofvariable x = r ( t − θ ) δ , we obtain that the Fisher information is I ( θ ) = (cid:90) θ + δθ (cid:0) − rδ (cid:1) ψ ( t ) + rδ ( t − θ ) d t = rδ (cid:90) r ψ (cid:0) θ + δr x (cid:1) + x d x, ψ ≡ λ , amounts to I ( θ ) = rδ ln (cid:18) λ + rλ (cid:19) . As for the change-point case (see Kutoyants (1984, 1998)), the likelihood normalizationrate can be chosen as ϕ n = 1 n . Note that in this case the processes Z n , n ∈ N , can be extended to the whole real lineso that their trajectories almost surely belong to the Skorokhod space D ( R ) of c`adl`agfunctions vanishing at ±∞ . The process Z n converge to the process Z (cid:63)a,b defined by Z (cid:63)a,b ( u ) = exp (cid:110) ln (cid:0) ab (cid:1) Y + ( u ) + ( b − a ) u (cid:111) , if u ≥ , exp (cid:110) ln (cid:0) ba (cid:1) Y − ( − u ) + ( b − a ) u (cid:111) , if u ≤ , (5)where a, b > Y + and Y − are independent Poisson processes on R + of constant intensities b and a respectively. Here, the convergence takes place in D ( R )equipped with the usual Skorokhod topology induced by the distance ρ ( f, g ) = inf ν (cid:20) sup u ∈ R (cid:12)(cid:12) f ( u ) − g (cid:0) ν ( u ) (cid:1)(cid:12)(cid:12) + sup u ∈ R (cid:12)(cid:12) u − ν ( u ) (cid:12)(cid:12)(cid:21) , where the inf is taken over all continuous one-to-one mappings ν : R −→ R .Then, the MLE and the BEs (for any continuous strictly positive prior density q ) areconsistent, converge at rate 1 /n : n (cid:0) ˆ θ n − θ (cid:1) = ⇒ ξ a,b and n (cid:0) ˜ θ n − θ (cid:1) = ⇒ ζ a,b , where ξ a,b = argsup u ∈ R Z (cid:63)a,b ( u )and ζ a,b = (cid:82) u ∈ R uZ (cid:63)a,b ( u ) d u (cid:82) u ∈ R Z (cid:63)a,b ( u ) d u , (6)we have the convergence of polynomial moments, and the BEs are asymptotically efficient.Note also that in the case of the intensity function given by (3), we have a = ψ ( θ )and b = ψ ( θ ) + r , which, in the particular case ψ ≡ λ , amounts to a = λ and b = λ + r .Finally, let us note that for our model, the trajectories of the (extended to the wholereal line) processes Z n , n ∈ N , almost surely belong to the space C ( R ). We will see laterin this paper that in the case nδ n → + ∞ , the trajectories of the limit process also belongto C ( R ), and the convergence takes place in this space equipped with the usual sup norm.However, this is not the case when nδ n →
0. In this case, the trajectories of the limitprocess must be discontinuous (cid:0) belong to D ( R ) \ C ( R ) (cid:1) , and hence the convergence cannot take place neither in the topology induced by the sup norm, nor in the usual Skorokhodtopology. 8 Main results
It turns out that the asymptotic behavior of our model depends on the rate of convergenceof δ n to zero. More precisely, there are three different cases: nδ n −−−−→ n → + ∞ + ∞ , (7) nδ n −−−−→ n → + ∞ nδ n −−−−→ n → + ∞ c > . (9)In this paper, we limit ourselves to the study of the MLE and of the BEs in the case (7),which we call slow case , and to the study of the BEs in the case (8), which we call fastcase .The study of the MLE in the fast case is more complicated, due to the already mentionedfact that in this case the convergence of the normalized likelihood ratio can not take placeneither in the topology induced by the sup norm, nor in the usual Skorokhod topology. So,the MLE in the fast case, as well as the case (9), will be considered in future works. Slow case
In order to study the behavior of the MLE and of the BEs of θ in the slow case, we choosethe likelihood normalization rate ϕ n = (cid:114) δ n n , and we denote F = F ( θ ) = r ln (cid:16) ψ ( θ ) + rψ ( θ ) (cid:17) . We also recall the random process Z ◦ F defined by (4) and introduce the random variable ζ F = ∆ F F = argsup u ∈ R Z ◦ F ( u ) = (cid:82) u ∈ R uZ ◦ F ( u ) d u (cid:82) u ∈ R Z ◦ F ( u ) d u ∼ N (cid:16) , F (cid:17) . Note that the rate ϕ n goes to zero faster than 1 / √ n (the classic rate of the regularcase) and, as it follows from (7), slower than 1 /n (the rate of the change-point case).Now we can state the following theorem giving a H´ajek-Le Cam lower bound on therisk of all the estimators. Theorem 1.
Suppose nδ n → + ∞ . Then, for all θ ∈ Θ , we have lim δ → lim n → + ∞ inf ¯ θ n sup | θ − θ | <δ ϕ − n E ( n ) θ (cid:0) ¯ θ n − θ (cid:1) ≥ F ( θ ) , where the inf is taken over all possible estimators ¯ θ n of the parameter θ . ϕ n = (cid:112) δ n /n ). More precisely, we have the following lemma, whichwill be proved in Section 4 by checking the conditions of Theorem 2.1 of Kutoyants (1998)and applying it. Lemma 1.
Suppose nδ n → + ∞ . Then, the normalized (using the rate ϕ n = (cid:112) δ n /n )likelihood ratio Z n ( u ) = d P θ + uϕn ( X ( n ) )d P θ , u ∈ U n , admits the representation Z n ( u ) = exp (cid:26) u ∆ n − u F + ε n ( θ, u ) (cid:27) , where ∆ n = ⇒ N (0 , F ) is given by ∆ n ( θ ) = − r √ nδ n n (cid:88) j =1 (cid:90) θ + δ n θ ψ ( θ ) + rδ n ( t − θ ) d X j ( t ) + r (cid:112) nδ n , and ε n ( θ, u ) converges to zero in probability. As it is usual in LAN situations, Theorem 1 allows us to introduce the following defi-nition of H´ajek-Le Cam efficiency.
Definition 1.
Suppose nδ n → + ∞ . We say that an estimator θ ∗ n of the parameter θ isasymptotically efficient if, for all θ ∈ Θ , we have lim δ → lim n → + ∞ sup | θ − θ | <δ ϕ − n E ( n ) θ (cid:0) θ ∗ n − θ (cid:1) = 1 F ( θ ) . Finally, the asymptotic properties of the MLE and of the BEs are given by the followingtheorem.
Theorem 2.
Suppose nδ n → + ∞ . Then, the MLE ˆ θ n and, for any continuous and strictlypositive prior density q , the BE ˜ θ n have the following proprieties: • ˆ θ n and ˜ θ n are consistent, • ˆ θ n and ˜ θ n are asymptotically normal with rate ϕ n = (cid:112) δ n /n and limit variance /F ,that is: ϕ − n (cid:0) ˆ θ n − θ (cid:1) = ⇒ ζ F and ϕ − n (cid:0) ˜ θ n − θ (cid:1) = ⇒ ζ F , • we have the convergence of polynomial moments, that is, for any p > , we have lim n → + ∞ ϕ − pn E ( n ) θ (cid:12)(cid:12) ˆ θ n − θ (cid:12)(cid:12) p = E | ζ F | p and lim n → + ∞ ϕ − pn E ( n ) θ (cid:12)(cid:12) ˜ θ n − θ (cid:12)(cid:12) p = E | ζ F | p , • ˆ θ n and ˜ θ n are asymptotically efficient. Note that the LAN property yields, in particular, the convergence of finite-dimensionaldistributions of the normalized likelihood ratio process Z n to those of the process Z ◦ F . So,in order to prove Theorem 2, it is sufficient to establish two additional lemmas (which willbe done in Section 4) and apply Theorems 1.10.1 and 1.10.2 of Ibragimov and Khasmin-skii (1981). 10 ast case In order to study the behavior of Bayesian estimators of θ in the fast case, we choose thelikelihood normalization rate ϕ n = 1 n , and we denote a = a ( θ ) = ψ ( θ ) and b = b ( θ ) = ψ ( θ ) + r. We also recall the random process Z (cid:63)a,b defined by (5) and the random variable ζ a,b definedby (6).Now we can state the following theorem giving a H´ajek-Le Cam type lower bound onthe risk of all the estimators. Theorem 3.
Suppose nδ n → . Then, for all θ ∈ Θ , we have lim δ → lim n → + ∞ inf ¯ θ n sup | θ − θ | <δ n E ( n ) θ (¯ θ n − θ ) ≥ E ζ a ( θ ) ,b ( θ ) , 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 2.
Suppose nδ n → . We say that an estimator θ ∗ n of the parameter θ isasymptotically efficient if, for all θ ∈ Θ , we have lim δ → lim n → + ∞ sup | θ − θ | <δ n E ( n ) θ ( θ ∗ n − θ ) = E ζ a ( θ ) ,b ( θ ) . Finally, the asymptotic properties of the BEs are given by the following theorem.
Theorem 4.
Suppose nδ n → . Then, for any continuous and strictly positive priordensity q , the BE ˜ θ n has the following proprieties: • ˜ θ n is consistent, • ˜ θ n converges at rate ϕ n = 1 /n and its limit law is that of the random variable ζ a,b ,that is: n (cid:0) ˜ θ n − θ (cid:1) = ⇒ ζ a,b , • we have the convergence of polynomial moments, that is, for any p > , we have lim n → + ∞ n p E ( n ) θ (cid:12)(cid:12) ˜ θ n − θ (cid:12)(cid:12) p = E | ζ a,b | p , • ˜ θ n is asymptotically efficient. Z n to the limit likelihood ratio Z (cid:63)a,b can not take place neither inthe topology induced by the sup norm, nor in the usual Skorokhod topology. Nevertheless,the convergence in a functional space is only needed for the properties of the MLE; inorder to obtain the lower bound and the properties of the BEs, it is sufficient to checkthe convergence of finite-dimensional distributions together with two additional lemmas(which will be established in Section 4) and apply Theorems 1.9.1 and 1.10.2 of Ibragimovand Khasminskii (1981). In order to simplify the exposition and make the ideas of the proofs clearer, we presentthem in the particular case ψ ≡ λ . Note that as in singular problems all the informationusually comes from the vicinity of the singularity, this is not a real loss of generality.Moreover, the given proofs can be easily extended to the general case (some details aregiven where necessary). Slow case
As we already explained above, Lemma 1 will be proved by applying Theorem 2.1 ofKutoyants (1998). So, we need to check the conditions of this theorem.For this, remind that our model of observation is equivalent to observing a singlerealization on [0 , τ ] of a Poisson process Y ( n ) = (cid:0) Y ( n ) ( t ) , ≤ t ≤ τ (cid:1) of intensity func-tion Λ ( n ) θ = n λ ( n ) θ . The process Y ( n ) can be defined, for example, by Y ( n ) ( t ) = (cid:80) nj =1 X j ( t ).Denote S n ( θ , θ , t ) = Λ ( n ) θ ( t )Λ ( n ) θ ( t )for all n ∈ N , t ∈ [0 , τ ] and θ , θ ∈ Θ.The conditions of Theorem 2.1 of Kutoyants (1998) are now the following. ( A ) For all n ∈ N , the intensity measures corresponding to (having them as Radon-Nikodym derivatives) the intensity functions Λ ( n ) θ , θ ∈ Θ, are equivalent. ( A ) For all n ∈ N , there exist some function q n ( θ, t ) such that Q n ( θ ) = (cid:90) τ q n ( θ, t ) Λ ( n ) θ ( t ) d t is positive for all θ ∈ Θ and, for any (cid:15) >
0, it holds (cid:90) τ (cid:12)(cid:12)(cid:12) Q − n ( θ ) q n ( θ, t ) (cid:12)(cid:12)(cid:12) (cid:8)(cid:12)(cid:12) Q − n ( θ ) q n ( θ,t ) (cid:12)(cid:12) >(cid:15) (cid:9) Λ ( n ) θ ( t ) d t −−−−→ n → + ∞ . A ) Let θ u = θ + u Q − n ( θ ) and U (cid:48) n = { u : θ u ∈ Θ } . For any u ∈ R , we have u ∈ U (cid:48) n for n sufficently large, and it holds (cid:90) τ (cid:104) ln S n ( θ u , θ, t ) − u Q − n ( θ ) q n ( θ, t ) (cid:105) Λ ( n ) θ ( t ) d t −−−−→ n → + ∞ (cid:90) τ (cid:20) S n ( θ u , θ, t ) − − ln S n ( θ u , θ, t ) − (cid:16) u Q − n ( θ ) q n ( θ, t ) (cid:17) (cid:21) Λ ( n ) θ ( t ) d t −−−−→ n → + ∞ . (11)Since the intensity functions Λ ( n ) θ , θ ∈ Θ, are strictly positive, the condition ( A ) istrivially verified.Now, in order to prove the conditions ( A ) and ( A ), we put q n ( θ, t ) = − rδ n λ + rδ n ( t − θ ) [ θ,θ + δ n ] ( t ) , and so Q n ( θ ) = n (cid:90) θ + δ n θ (cid:0) − rδ n (cid:1) λ + rδ n ( t − θ ) d t = r nδ n (cid:90) λ + rλ x d x = r ln (cid:16) λ + rλ (cid:17) nδ n = F ϕ − n , where we used the change of variable x = λ + rδ n ( t − θ ) . Therefore, we have Q − n ( θ ) = 1 (cid:113) r ln (cid:0) λ + rλ (cid:1) (cid:114) δ n n = ϕ n √ F = γϕ n , where we denoted γ = 1 √ F .
Proof of ( A ) . Let θ ∈ Θ and (cid:15) >
0. Denoting F n = (cid:90) τ (cid:12)(cid:12)(cid:12) Q − n ( θ ) q n ( θ, t ) (cid:12)(cid:12)(cid:12) (cid:8)(cid:12)(cid:12) Q − n ( θ ) q n ( θ,t ) (cid:12)(cid:12) >(cid:15) (cid:9) Λ ( n ) θ ( t ) d t, we need to prove that F n −−−−→ n → + ∞ .
13e can write F n = ( rγ ) δ n (cid:90) θ + δ n θ λ + rδ n ( t − θ ) (cid:110)(cid:12)(cid:12) − √ nδn rγλ rδn ( t − θ ) (cid:12)(cid:12) >(cid:15) (cid:111) d t = 1ln (cid:0) λ + rλ (cid:1) (cid:90) λ + rλ x (cid:8) | r | γ(cid:15) √ nδn >x (cid:9) d x, where we used again the change of variable x = λ + rδ n ( t − θ ) . Since nδ n −−−−→ n → + ∞ + ∞ , we have F n = 0for n sufficiently large, and so we get the condition ( A ). Proof of ( A ) . First of all, let us note that U (cid:48) n = ] ϕ − n γ − ( α − θ ) , ϕ − n γ − ( β − θ )[, and as ϕ n →
0, we get U (cid:48) n ↑ R . Thus, for any u ∈ R , we have u ∈ U (cid:48) n for n sufficiently large.Further we consider the case where u > r > γuϕ n < δ n for n sufficiently large. Wepresent in Figure 3 the functions λ θ and λ θ u , where θ u = θ + u Q − n ( θ ) = θ + γuϕ n , with u, r > n (cid:29) G n = (cid:90) τ (cid:104) ln S n ( θ u , θ, t ) − u Q − n ( θ ) q n ( θ, t ) (cid:105) Λ ( n ) θ ( t ) d t. We can write G n = n (cid:90) θ + γuϕ n θ (cid:16) λ + rδ n ( t − θ ) (cid:17)(cid:20) ln (cid:18) λ λ + rδ n ( t − θ ) (cid:19) + ϕ n δ n γruλ + rδ n ( t − θ ) (cid:21) d t + n (cid:90) θ + δ n θ + γuϕ n (cid:16) λ + rδ n ( t − θ ) (cid:17)(cid:20) ln (cid:18) λ + rδ n ( t − θ − γuϕ n ) λ + rδ n ( t − θ ) (cid:19) + ϕ n δ n γruλ + rδ n ( t − θ ) (cid:21) d t + n (cid:90) θ + δ n + γuϕ n θ + δ n ( λ + r ) ln (cid:32) λ + rδ n ( t − θ − γuϕ n ) λ + r (cid:33) d t = B + B + B with evident notations, and so it is sufficient to show that B , B and B go to zero.14 tq q + g u j n q + d n q + g u j n + d n l l + r l q l q u Figure 3: λ θ and λ θ u with u, r > n (cid:29) B −−−−→ n → + ∞ . Using the change of variable x = t − θγuϕ n , we obtain B = γunϕ n (cid:90) (cid:18) λ + γru ϕ n δ n x (cid:19)(cid:20) ln (cid:18) γruλ ϕ n δ n x (cid:19) − ϕ n δ n γruλ + γru ϕ n δ n x (cid:21) d x. Since, for 0 ≤ x ≤
1, we have λ + γru ϕ n δ n x ≤ λ + r and (cid:20) ln (cid:18) γruλ ϕ n δ n x (cid:19) − ϕ n δ n γruλ + γru ϕ n δ n x (cid:21) ≤ ln (cid:18) γruλ ϕ n δ n x (cid:19) + (cid:20) ϕ n δ n γruλ + γru ϕ n δ n x (cid:21) ≤ (cid:20) γruλ ϕ n δ n x (cid:21) + (cid:20) ϕ n δ n γruλ + γru ϕ n δ n x (cid:21) , ≤ γru ) λ ϕ n δ n ,
15t comes B ≤ λ + r )( γu ) r λ nϕ n δ n . Finally, as nϕ n δ n = 1 √ nδ n −−−−→ n → + ∞ , we get B −−−−→ n → + ∞ . Proceeding similarly, it is not difficult to verify that B and B go to zero as well, andso (10) is proved.Now, in order to prove (11), we denote H n = (cid:90) τ h n ( t ) d t, where h n ( t ) = (cid:20) S n ( θ u , θ, t ) − − ln S n ( θ u , θ, t ) − (cid:16) u Q − n ( θ ) q n ( θ, t ) (cid:17) (cid:21) Λ ( n ) θ ( t ) . We have | H n | ≤ (cid:90) θ + γuϕ n θ (cid:12)(cid:12) h n ( t ) (cid:12)(cid:12) d t + (cid:90) θ + δ n θ + γuϕ n (cid:12)(cid:12) h n ( t ) (cid:12)(cid:12) d t + (cid:90) θ + γuϕ n + δ n θ + δ n (cid:12)(cid:12) h n ( t ) (cid:12)(cid:12) d t = D + D + D with evident notations, and so it is sufficient to show that D , D and D go to zero.Let us verify, for example, that D −−−−→ n → + ∞ . Using the change of variable x = t − θγuϕ n , we obtain D = n (cid:90) θ + γuϕ n θ (cid:16) λ + rδ n ( t − θ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ λ + rδ n ( t − θ ) − − ln (cid:18) λ λ + rδ n ( t − θ ) (cid:19) − ( γru ) ϕ n δ n (cid:0) λ + rδ n ( t − θ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d t γunϕ n (cid:90) ( λ + r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ λ + γru ϕ n δ n x − (cid:18) γruλ ϕ n δ n x (cid:19) − ( γru ) ϕ n δ n (cid:0) λ + γru ϕ n δ n x (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x = γunϕ n (cid:90) ( λ + r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) γruλ ϕ n δ n x (cid:19) − γru ϕ n δ n xλ + γru ϕ n δ n x − ( γru ) ϕ n δ n (cid:0) λ + γru ϕ n δ n x (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x. Since, for 0 ≤ x ≤
1, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) γruλ ϕ n δ n x (cid:19) − γru ϕ n δ n xλ + γru ϕ n δ n x − ( γru ) ϕ n δ n (cid:0) λ + γru ϕ n δ n x (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) γruλ ϕ n δ n x (cid:19) − γruλ ϕ n δ n x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γruλ ϕ n δ n x (cid:18) − λ λ + γru ϕ n δ n x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ( γru ) ϕ n δ n (cid:0) λ + γru ϕ n δ n x (cid:1) ≤ (cid:18) γruλ ϕ n δ n x (cid:19) + ( γru ) λ ϕ n δ n x λ + γru ϕ n δ n x + ( γru ) ϕ n δ n (cid:0) λ + γru ϕ n δ n x (cid:1) ≤ γru ) λ ϕ n δ n , it comes D ≤ λ + r )( γu ) r λ nϕ n δ n −−−−→ n → + ∞ . Proceeding similarly, it is not difficult to verify that D and D go to zero as well, andso (11) is proved.Now, we can apply Theorem 2.1 of Kutoyants (1998), which yields that the family { P ( n ) θ , θ ∈ Θ } is LAN with rate ϕ (cid:48) n = Q − n ( θ ) = γϕ n . More precisely, for all u ∈ U (cid:48) n , we have Z (cid:48) n ( u ) = d P θ + uϕ (cid:48) n (cid:0) Y ( n ) (cid:1) d P θ = exp (cid:110) u ∆ (cid:48) n ( θ ) − u ε (cid:48) n ( θ, u ) (cid:111) , (cid:48) n = ⇒ N (0 ,
1) is given by∆ (cid:48) n ( θ ) = Q − n ( θ ) (cid:90) τ q n ( θ, t ) (cid:2) d Y ( n ) ( t ) − Λ ( n ) θ ( t ) d t (cid:3) = γ (cid:114) δ n n (cid:90) θ + δ n θ − rδ n λ ( n ) θ ( t ) (cid:2) d Y ( n ) ( t ) − nλ ( n ) θ ( t ) d t (cid:3) = − rγ √ nδ n (cid:90) θ + δ n θ λ + rδ n ( t − θ ) d Y ( n ) ( t ) + rγ (cid:112) nδ n , and ε (cid:48) n ( θ, u ) converges to zero in probability.Finally, it is clear that we can equivalently restate the LAN property using the rate ϕ n = Q − n ( θ ) γ = (cid:114) δ n n and the observations X ( n ) . In this case, for all u ∈ U n , we have Z n ( u ) = d P θ + uϕ n (cid:0) X ( n ) (cid:1) d P θ = exp (cid:110) u ∆ n ( θ ) − u F + ε n ( θ, u ) (cid:111) , where ∆ n = ⇒ N (0 , F ) is given by∆ n ( θ ) = ∆ (cid:48) n ( θ ) γ = − r √ nδ n n (cid:88) j =1 (cid:90) θ + δ n θ λ + rδ n ( t − θ ) d X j ( t ) + r (cid:112) nδ n , and ε n ( θ, u ) = ε (cid:48) n (cid:0) θ, uγ (cid:1) converges to zero in probability.To wrap up this part, let us note that it is not very difficult to adapt the proofs ofthe conditions ( A ) and ( A ) to the case where ψ is any strictly positive continuous (notnecessarily constant) function on [0 , τ ]. In this case, taking q n ( θ, t ) = − rδ n ψ ( θ ) + rδ n ( t − θ ) [ θ,θ + δ n ] ( t ) , we get Q n ( θ ) = (cid:90) τ q n ( θ, t )Λ ( n ) θ ( t ) d t = nr δ n (cid:90) θ + δ n θ ψ ( t ) + rδ n ( t − θ ) (cid:16) ψ ( θ ) + rδ n ( t − θ ) (cid:17) d t = nrδ n (cid:90) r ψ (cid:0) θ + δ n r x (cid:1) + x ( ψ ( θ ) + x ) d x F ϕ − n + rϕ − n (cid:90) r ψ (cid:0) θ + δ n r x (cid:1) − ψ ( θ )( ψ ( θ ) + x ) d x = F ϕ − n (cid:0) o (1) (cid:1) . Note also, that since the function ψ is continuous and strictly positive on [0 , τ ], it admitsa minimum m > M >
0, which can replace λ in different estimatesused in the proofs of the condition ( A ). For the case r <
0, it is equally important toremind that we supposed that r > − m , and hence M + r > m + r > Z n to those ofthe process Z ◦ F follows from the already established LAN property, it remains to prove thefollowing two lemmas. Lemma 2.
Suppose nδ n → + ∞ . Then, there exists a constant C > , such that for n sufficiently large, we have E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ C | u − v | (12) for all u, v ∈ U n and θ ∈ Θ . Lemma 3.
Suppose nδ n → + ∞ . Then, there exists a constant κ > , such that for n sufficiently large, we have E Z / n ( u ) ≤ exp (cid:8) − κ min {| u | , u } (cid:9) (13) for all u ∈ U n and θ ∈ Θ .Proof of Lemma 2. First of all, let us note that, as for | u − v | ≥ E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ ≤ | u − v | , it is sufficient to consider the case | u − v | ≤
1. Moreover, without loss of generality, we cansuppose that u < v .Further we consider the case r > r < E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ n (cid:90) τ (cid:16)(cid:113) λ θ + uϕ n ( t ) − (cid:113) λ θ + vϕ n ( t ) (cid:17) d t = n (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) (cid:1) (cid:0)(cid:112) λ θ + uϕ n ( t ) + (cid:112) λ θ + vϕ n ( t ) (cid:1) d t n λ (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) (cid:1) d t = n λ (cid:90) τ f n ( t ) d t, where we denoted f n ( t ) = (cid:0) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) (cid:1) .Taking into account that δ n /ϕ n = √ nδ n → + ∞ , for n sufficiently large (such that nδ n ≥ v − u ) ϕ n ≤ ϕ n ≤ δ n , and hence vϕ n ≤ uϕ n + δ n . Therefore, we canwrite (cid:90) τ f n ( t ) d t = (cid:90) θ + vϕ n θ + uϕ n f n ( t ) d t + (cid:90) θ + uϕ n + δ n θ + vϕ n f n ( t ) d t + (cid:90) θ + vϕ n + δ n θ + uϕ n + δ n f n ( t ) d t = E + E + E with evident notations.For E , we have E = (cid:90) θ + vϕ n θ + uϕ n (cid:16) rδ n ( t − θ − uϕ n ) (cid:17) d t = r δ n ( vϕ n − uϕ n ) r ϕ n δ n ( v − u ) , and proceeding similarly, we get E = r ϕ n δ n ( v − u ) − r ϕ n δ n ( v − u ) and E = r ϕ n δ n ( v − u ) . Thus, using the fact that nϕ n /δ n = 1, we have E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ r λ nϕ n δ n ( v − u ) − r λ nϕ n δ n ( v − u ) ≤ r λ ( v − u ) , and so the inequality (12) is proved with C = max (cid:8) , r λ (cid:9) . Proof of Lemma 3.
We consider the case where u > r > E Z / n ( u ) = exp (cid:26) − n (cid:90) τ (cid:16)(cid:113) λ θ + uϕ n ( t ) − (cid:112) λ θ ( t ) (cid:17) d t (cid:27) = exp (cid:26) − n (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ ( t ) (cid:1) (cid:0)(cid:112) λ θ + uϕ n ( t ) + (cid:112) λ θ ( t ) (cid:1) d t (cid:27) ≤ exp (cid:26) − n λ + r ) (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ ( t ) (cid:1) d t (cid:27) = exp (cid:26) − n λ + r ) (cid:90) τ g n ( t ) d t (cid:27) , g n ( t ) = (cid:0) λ θ + uϕ n ( t ) − λ θ ( t ) (cid:1) .Now we treat separately two cases: u ≤ δ n /ϕ n and u ≥ δ n /ϕ n . In the first case, thesituation is similar to that of Figure 3, and so we obtain (cid:90) τ g n ( t ) d t = (cid:90) θ + uϕ n θ g n ( t ) d t + (cid:90) θ + δ n θ + uϕ n g n ( t ) d t + (cid:90) θ + uϕ n + δ n θ + δ n g n ( t ) d t = J + J + J with evident notations.For J , we have J = (cid:90) θ + uϕ n θ (cid:16) rδ n ( t − θ ) (cid:17) d t = r ϕ n δ n u , and proceeding similarly, we obtain J = r ϕ n δ n u − r ϕ n δ n u and J = r ϕ n δ n u . Thus, using the fact that u ≤ δ n /ϕ n , we get E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) nϕ n δ n u + r λ + r ) nϕ n δ n u (cid:27) ≤ exp (cid:26) − r λ + r ) nϕ n δ n u (cid:27) ≤ exp (cid:26) − r λ + r ) nϕ n δ n min { u, u } (cid:27) , (14)and recalling that nϕ n /δ n = 1, we conclude that E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) min { u, u } (cid:27) . In the second case ( u ≥ δ n /ϕ n ), the situation is that of Figure 4, and so we obtain (cid:90) τ g n ( t ) d t = (cid:90) θ + δ n θ g n ( t ) d t + (cid:90) θ + uϕ n θ + δ n g n ( t ) d t + (cid:90) θ + uϕ n + δ n θ + uϕ n g n ( t ) d t = J (cid:48) + J (cid:48) + J (cid:48) with evident notations.For J (cid:48) , we have J (cid:48) = (cid:90) θ + δ n θ (cid:16) rδ n ( t − θ ) (cid:17) d t = r δ n , and proceeding similarly, we obtain J (cid:48) = r uϕ n − r δ n and J (cid:48) = r δ n . δ n ≤ uϕ n , we get E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) nϕ n u + r λ + r ) nδ n (cid:27) ≤ exp (cid:26) − r λ + r ) nϕ n u (cid:27) ≤ exp (cid:26) − r λ + r ) nϕ n min { u, u } (cid:27) , (15)and taking into account that nϕ n = √ nδ n → + ∞ , for n sufficiently large (such that nδ n ≥ E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) min { u, u } (cid:27) , and so the inequality (13) is proved with κ = r λ + r ) .Note that we can easily adapt the proofs of Lemmas 2 and 3 to the case where ψ is anystrictly positive continuous (not necessarily constant) function on [0 , τ ], and so Theorem 2is proved. Fast case
As we already explained above, Theorems 3 and 4 can be proved by applying Theo-rems 1.9.1 and 1.10.2 of Ibragimov and Khasminskii (1981). So, it is sufficient to check theconditions of these theorems, that is, to prove the following three lemmas.
Lemma 4.
Suppose nδ n → . Then, the finite-dimensional distributions of the process Z n converge to those of the process Z (cid:63)a,b with a = ψ ( θ ) and b = ψ ( θ ) + r . Lemma 5.
Suppose nδ n → . Then, there exists a constant C > , such that E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ C | u − v | (16) for all n ∈ N , u, v ∈ U n and θ ∈ Θ . Lemma 6.
Suppose nδ n → . Then, there exists a constant κ > , such that for n sufficiently large, we have E Z / n ( u ) ≤ exp (cid:8) − κ min {| u | , u } (cid:9) (17) for all u ∈ U n and θ ∈ Θ . Before proving Lemmas 4–6, note that since we are in the fast case, for any u ∈ R , wehave uϕ n > δ n for n sufficiently large. We present in Figure 4 the functions λ θ and λ θ u ,where θ u = θ + uϕ n , with u, r > n (cid:29) tq q + d n q + u j n q + u j n + d n l l + r l q l q u Figure 4: λ θ and λ θ u with u, r > n (cid:29) Proof of Lemma 4.
We study the convergence of 2-dimensional distributions only (the con-vergence of d -dimensional distributions for d ≥ u, v ∈ R and consider the distribution of the vector (cid:0) ln Z n ( u ) , ln Z n ( v ) (cid:1) ,where n is sufficiently large, so that u, v ∈ U n . Its characteristic function is given, for all x, y ∈ R , by φ (cid:0) ln Z n ( u ) , ln Z n ( v ) (cid:1)(cid:0) x, y (cid:1) = E exp (cid:8) ix ln Z n ( u ) + iy ln Z n ( v ) (cid:9) = E exp (cid:40) ix n (cid:88) j =1 (cid:20)(cid:90) τ ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) − (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ ( t ) (cid:1) d t (cid:1)(cid:21) + iy n (cid:88) j =1 (cid:20)(cid:90) τ ln (cid:18) λ θ + vϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) − (cid:90) τ (cid:0) λ θ + vϕ n ( t ) − λ θ ( t ) (cid:1) d t (cid:1)(cid:21)(cid:41) = E exp (cid:40) ix n (cid:88) j =1 (cid:20)(cid:90) τ ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) + ruϕ n (cid:21) + iy n (cid:88) j =1 (cid:20)(cid:90) τ ln (cid:18) λ θ + vϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) + rvϕ n (cid:21)(cid:41) = exp (cid:8) nir ( ux + vy ) ϕ n (cid:9) × E exp (cid:40) ix n (cid:88) j =1 (cid:90) τ ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) + iy n (cid:88) j =1 (cid:90) τ ln (cid:18) λ θ + vϕ n ( t ) λ θ ( t ) (cid:19) d X j ( t ) (cid:41)
23 exp (cid:8) ir ( ux + vy ) (cid:9) × exp (cid:40) n (cid:90) τ (cid:32) exp (cid:26) ix ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) + iy ln (cid:18) λ θ + vϕ n ( t ) λ θ ( t ) (cid:19)(cid:27) − (cid:33) λ θ ( t ) d t (cid:41) = exp (cid:8) ir ( ux + vy ) (cid:9) exp (cid:26) n (cid:90) τ f n ( t ) d t (cid:27) with an evident notation.We consider the case where v > u ≥ r > n sufficiently large, we have δ n < uϕ n and uϕ n + δ n < vϕ n ,and so we can write (cid:90) τ f n ( t ) d t = (cid:90) θ + δ n θ f n ( t ) d t + (cid:90) θ + uϕ n θ + δ n f n ( t ) d t + (cid:90) θ + uϕ n + δ n θ + uϕ n f n ( t ) d t + (cid:90) θ + vϕ n θ + uϕ n + δ n f n ( t ) d t + (cid:90) θ + vϕ n + δ n θ + vϕ n f n ( t ) d t = (cid:88) j =1 I j with evident notations.For I , using the change of variable s = t − θδ n , we have I = (cid:90) θ + δ n θ f n ( t ) d t = (cid:90) θ + δ n θ (cid:32) exp (cid:26) ix ln (cid:18) λ θ + uϕ n ( t ) λ θ ( t ) (cid:19) + iy ln (cid:18) λ θ + vϕ n ( t ) λ θ ( t ) (cid:19)(cid:27) − (cid:33) λ θ ( t ) d t = (cid:90) θ + δ n θ (cid:32) exp (cid:26) i ( x + y ) ln (cid:18) λ λ + rδ n ( t − θ ) (cid:19)(cid:27) − (cid:33)(cid:18) λ + rδ n ( t − θ ) (cid:19) d t = δ n (cid:90) (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + rs (cid:17)(cid:111) − (cid:19) ( λ + rs ) d s = c δ n , where c is some constant, and proceeding similarly, we obtain I = c δ n and I = c δ n .For I , we have I = (cid:90) θ + uϕ n θ + δ n f n ( t ) d t = (cid:90) θ + uϕ n θ + δ n (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19) ( λ + r ) d t = (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19) ( λ + r )( uϕ n − δ n ) , I = (cid:18) exp (cid:110) iy ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19) ( λ + r ) (cid:0) ( v − u ) ϕ n − δ n (cid:1) . . Therefore, it comes (cid:90) τ f n ( t ) d t = Cδ n + (cid:20) ( v − u ) (cid:18) exp (cid:110) iy ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19) + u (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19)(cid:21) ( λ + r ) ϕ n , and hence, recalling that ϕ n = 1 /n , we getexp (cid:26) n (cid:90) τ f n ( t ) d t (cid:27) = exp { Cnδ n } exp (cid:26) u ( λ + r ) (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19)(cid:27) × exp (cid:26) ( v − u )( λ + r ) (cid:18) exp (cid:110) iy ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19)(cid:27) . Since nδ n −−−−→ n → + ∞ , we finally conclude that φ (cid:0) ln Z n ( u ) , ln Z n ( v ) (cid:1) ( x, y ) −−−−→ n → + ∞ exp { ir ( ux + vy ) }× exp (cid:26) u ( λ + r ) (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19)(cid:27) × exp (cid:26) ( v − u )( λ + r ) (cid:18) exp (cid:110) iy ln (cid:16) λ λ + r (cid:17)(cid:111) − (cid:19)(cid:27) . (18)Now, let us calculate, still for v > u ≥
0, the characteristic function of the vector (cid:0) ln Z (cid:63)a,b ( u ) , ln Z (cid:63)a,b ( v ) (cid:1) . For all x , y ∈ R , we have φ (cid:0) ln Z (cid:63)a,b ( u ) , ln Z (cid:63)a,b ( v ) (cid:1) ( x, y ) = E exp (cid:8) ix ln Z (cid:63)a,b ( u ) + iy ln Z (cid:63)a,b ( v ) (cid:9) = E exp (cid:26) ix (cid:18) ln (cid:16) ab (cid:17) Y + ( u ) + ( b − a ) u (cid:19) + iy (cid:18) ln (cid:16) ab (cid:17) Y + ( v ) + ( b − a ) v (cid:19)(cid:27) = exp { i ( b − a )( ux + vy ) }× E exp (cid:110) ix ln (cid:16) ab (cid:17) Y + ( u ) + iy ln (cid:16) ab (cid:17) Y + ( v ) (cid:111) . Y + is a Poison process and v > u ≥
0, we obtain E exp (cid:110) ix ln (cid:16) ab (cid:17) Y + ( u ) + iy ln (cid:16) ab (cid:17) Y + ( v ) (cid:111) = E exp (cid:110) i ( x + y ) ln (cid:16) ab (cid:17) Y + ( u ) (cid:111) E exp (cid:110) iy ln (cid:16) ab (cid:17)(cid:0) Y + ( v ) − Y + ( u ) (cid:1)(cid:111) = exp (cid:26) ub (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) ab (cid:17)(cid:111) − (cid:19)(cid:27) exp (cid:26) ( v − u ) b (cid:18) exp (cid:110) iy ln (cid:16) ab (cid:17)(cid:111) − (cid:19)(cid:27) . Therefore, we get φ (cid:0) ln Z (cid:63)a,b ( u ) , ln Z (cid:63)a,b ( v ) (cid:1) ( x, y ) = exp { i ( b − a )( ux + vy ) } exp (cid:26) ub (cid:18) exp (cid:110) i ( x + y ) ln (cid:16) ab (cid:17)(cid:111) − (cid:19)(cid:27) × exp (cid:26) ( v − u ) b (cid:18) exp (cid:110) iy ln (cid:16) ab (cid:17)(cid:111) − (cid:19)(cid:27) , which, taking a = ψ ( θ ) = λ and b = ψ ( θ ) + r = λ + r , is the same as the right hand sideof (18). This shows that (cid:0) ln Z n ( u ) , ln Z n ( v ) (cid:1) converge to (cid:0) ln Z (cid:63)λ ,λ + r ( u ) , ln Z (cid:63)λ ,λ + r ( v ) (cid:1) ,and hence the convergence of 2-dimensional distributions of Z n to those of Z (cid:63)λ ,λ + r isproved. Proof of Lemma 5.
We consider the case r > r < u < v . According toLemma 1.5 of Kutoyants (1998), we have E (cid:12)(cid:12) Z / n ( u ) − Z / n ( v ) (cid:12)(cid:12) ≤ n (cid:90) τ (cid:16)(cid:113) λ θ + uϕ n ( t ) − (cid:113) λ θ + vϕ n ( t ) (cid:17) d t. Since λ θ + uϕ n ( t ) ≥ λ θ + vϕ n ( t )and (cid:113) λ θ + uϕ n ( t ) − (cid:113) λ θ + vϕ n ( t ) ≤ (cid:113) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) , we obtain (cid:90) τ (cid:16)(cid:113) λ θ + uϕ n ( t ) − (cid:113) λ θ + vϕ n ( t ) (cid:17) d t ≤ (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) (cid:1) d t. By a simple area calculation, we get (cid:90) τ (cid:0) λ θ + uϕ n ( t ) − λ θ + vϕ n ( t ) (cid:1) d t = r ( v − u ) ϕ n , which, taking into account that ϕ n = 1 /n , yields the inequality (16) with C = r .26 roof of Lemma 6. We consider the case where u > r > u ≤ δ n /ϕ n and u ≥ δ n /ϕ n .In the first case, we have already shown (see (14), the proof of which is valid also in thefast case) that E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) nϕ n δ n min { u, u } (cid:27) . Thus, taking into account that nϕ n δ n = nδ n → + ∞ , for n sufficiently large (such that nδ n ≤ E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) min { u, u } (cid:27) . In the second case ( uϕ n ≥ δ n ), we have already shown (see (15), the proof of which isvalid also in the fast case) that E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) nϕ n min { u, u } (cid:27) . Thus, recalling that ϕ n = 1 /n , we conclude again that E Z / n ( u ) ≤ exp (cid:26) − r λ + r ) min { u, u } (cid:27) , and so the inequality (17) is proved with κ = r λ + r ) .Note that we can easily adapt the proofs of Lemmas 4–6 to the case where ψ is anystrictly positive continuous (not necessarily constant) function on [0 , τ ], and so Theorems 3and 4 are proved. Acknowledgments
This research was financially supported by RFBR and CNRS (project 20–51–15001). Theauthors equally acknowledge support from the Labex CEMPI (ANR–11–LABX–0007–01).
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