Delphine Blanke
email: delphine.blanke@univ-avignon.fr

Denis Bosq
email: denis.bosq@upmc.fr

DETECTING AND ESTIMATING INTENSITY OF JUMPS FOR DISCRETELY OBSERVED ARMAD(1, 1) PROCESSES

Keywords: 1991 Mathematics Subject Classification. 62M, 60F17. Key words and phrases. Functional linear processes, ARMAD(1, 1) processes, Jumps, Estimation of intensity, Discrete data

We consider n equidistributed random functions, defined on [0,1], and admitting fixed or random jumps, the context being D[0, 1]-valued ARMA (1,1) processes. We begin with properties of ARMAD(1, 1) processes. Next, different scenarios are considered: fixed instants with a given but unknown probability of jumps (the deterministic case), random instants with ordered intensities (the random case), and random instants with non ordered intensities (the completely random case). By using discrete data and for each scenario, we identify the instants of jumps, whose number is either random or fixed, and then estimate their intensity.

1. Introduction 1.1. There is an abundant literature concerning functional data analysis (FDA) and prediction of stochastic processes in infinite dimensional spaces. In particular, the books by [START_REF] Ramsay | Functional Data Analysis[END_REF]; [START_REF] Ferraty | Nonparametric functional data analysis[END_REF]; Ferraty and Romain (2011); [START_REF] Horváth | Inference for functional data with applications[END_REF] and the recent book edited by [START_REF] Bongiorno | Contributions in infinitedimensional statistics and related topics[END_REF] contain interesting theoretical and practical results. See also [START_REF] Bosq | Linear processes in function spaces[END_REF]; [START_REF] Bosq | Prediction and inference in large dimensions[END_REF]. In general, X takes its value in L 2 = L 2 ([0, h]) or in C = C([0, h]), but, in some situations, one may consider that a jump does exist if there is a large peak: see, for example the annual sediment in Bongiorno et al. (2014, p.8). Thus, it is perhaps more natural to consider the space D = D([0, h]) which is càdlàg and equipped with the Skorohod metric d • (see Billingsley, 1999, p.125) : with that metric, D becomes a separable complete metric space. Note that this metric is not easy to compute. In this paper, we consider càdlàg processes from a functional point of view: by this way, we work in the context of FDA with jumps. 1.2. Works dedicated to jumps in stochastic processes appear very often: actually, there are more than 1200 papers concerning them. Thus, we may only give recent and limited references. For example, processes with jumps are widely used in finance: we may refer to [START_REF] Cont | Financial modelling with jump processes[END_REF]; Tankov and Voltchkova (2009); Jeanblanc et al. (2009, part 2), El Karoui and Gobet (2012, ch.10); [START_REF] Privault | Stochastic finance. Chapman & Hall/CRC Financial Mathematics Series[END_REF], ... ; but applications can also be found in fields as varied as the environment, medicine, reliability, ... see e.g. [START_REF] Guy | Approximation of epidemic models by diffusion processes and their statistical inference[END_REF]; [START_REF] Barndorff-Nielsen | Modelling electricity futures by ambit fields[END_REF]; [START_REF] Chiquet | Dynamical systems with semi-Markovian perturbations and their use in structural reliability[END_REF]; [START_REF] Borisov | Analysis and estimation of the states of special Markov jump processes. II. Optimal filtering in the presence of Wiener noise[END_REF], ... Many mathematical models have been proposed and studied [START_REF] Djebali | Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces[END_REF][START_REF] Koroliuk | Poisson approximation of processes with locally independent increments with Markov switching[END_REF][START_REF] Guyon | Linear diffusion with stationary switching regime[END_REF][START_REF] De Saporta | Tail of a linear diffusion with Markov switching[END_REF], and statistical estimation appears e.g. in [START_REF] Comte | Nonparametric density estimation in compound Poisson processes using convolution power estimators[END_REF]; [START_REF] Clément | Asymptotic lower bounds in estimating jumps[END_REF]; [START_REF] Duval | Density estimation for compound Poisson processes from discrete data[END_REF],... Note that the pioneer paper concerning jumps appears in Paul [START_REF] Lévy | Fonctions aléatoires à corrélation linéaire[END_REF]. Other references of interest will appear below. 1.3. Now, here and in the books quoted in Section 1.1, our purpose is somewhat different since we want to observe a process over a sequence of time intervals. More precisely, let (ξ t , t ∈ R) be a real measurable continuous time process. We put

X n (t) = ξ (n-1)h+t , 0 ≤ t ≤ h, n ∈ Z (1.1)
where h > 0 is a time interval. The process may contain some jumps and we envisage to detect them and to estimate intensity of jumps, given the data X 1 , . . . , X n . Another motivation should be prediction of X n+1 over the time interval [nh, (n+ 1)h]. One way to predict X n+1 would be to treat continuous time and jumps separately (see [START_REF] Shimizu | Threshold selection in jump-discriminant filter for discretely observed jump processes[END_REF]; [START_REF] Aït-Sahalia | Fisher's information for discretely sampled Lévy processes[END_REF]; [START_REF] Shimizu | Estimation of parameters for diffusion processes with jumps from discrete observations[END_REF], ...). As an example, consider the functional autoregressive process of order 1 (ARD(1)):

X n+1 (t) = ρ(X n )(t) + Z n+1 (t), 0 ≤ t ≤ h, n ∈ Z,
where ρ is a continuous linear operator with respect to the sup-norm. Then, in order to separate the continuous part from the jump's part, we may suppose that ρ(D) ⊂ C. That condition is satisfied by the Ornstein-Uhlenbeck process driven by a Levy process, cf Example 2.1. Another classical example is given by: ρ r (x)(t) = h 0 r(s, t)x(s) ds, 0 ≤ t ≤ h, x ∈ D where r satisfies Example 2.2, see also [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF]; [START_REF] Horváth | Inference for functional data with applications[END_REF],... Thus, ρ r (x) ∈ C. Finally, the condition ρ(D) ⊂ C seems quite standard and characterizes the unpredictability of jumps by confining them in the innovation process. Now, the best probabilistic predictor of X n+1 is ρ(X n ) and it can be approximated by using an estimator of ρ. An exponential rate is obtained in Bosq (2000, p.222-235), when the detector and intensity of jumps appear in the current paper. One direction (currently under development) will consist in combining the two approaches to improve the prediction. 1.4. A more general model should be the ARMAD(p, p) process defined by

X n -ρ 1 (X n-1 ) -• • • -ρ p (X n-p ) = Z n -ρ ′ 1 (Z n-1 ) -• • • -ρ ′ p (Z n-p )
, n ∈ Z, where X n and Z n are D-valued and where ρ j , ρ ′ j , j, j ′ = 1, . . . , p are continuous linear operators with respect to the sup-norm. In order to study this process, it should be possible to work in the space D([0, h] p ) (cf [START_REF] Kurchenko | Convergence of a sequence of random fields in the space D([0, 1] d )[END_REF]. Note that if ρ j , ρ ′ j , j, j ′ = 1, . . . , p are C-valued, X n and Z n have again the same jumps. Now, since this model is difficult to handle, and in order to simplify the exposition, we take p = 1 and write

X n -ρ(X n-1 ) = Z n -ρ ′ (Z n-1 ), n ∈ Z,
note that, Z n-1 may be replaced with an exogenous variable (see for example [START_REF] Goia | A functional linear model for time series prediction with exogenous variables[END_REF]. 1.5. We now give some practical examples of jumps over time intervals:

-a patient's electrocardiogram at each minute [START_REF] Nason | A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series[END_REF][START_REF] Preston | Discovering arbitrary event types in time series[END_REF][START_REF] Marion | Comparaison des modèles ARH(1) et ARHD(1) sur des données physiologiques[END_REF]; -the temperature day by day [START_REF] Torgovitski | A Darling-Erdős-type CUSUM-procedure for functional data[END_REF]; -El Niño southern oscillation (ENSO): a prediction over one year shows a jump in may [START_REF] Besse | Autoregressive forecasting of some climatic variation[END_REF]; -wave amplitude [START_REF] Tanushev | Superpositions and higher order Gaussian beams[END_REF]; -pollution day by day [START_REF] Horváth | Inference for functional data with applications[END_REF]; -credit cards transaction and its prediction [START_REF] Horváth | Inference for functional data with applications[END_REF]; -another example is electricity consumption: it admits a jump early in the morning and in the evening (see [START_REF] Antoniadis | Prévision d'un processus à valeurs fonctionnelles en présence de non stationnarités. Application à la consommation d'électricité[END_REF][START_REF] El Hajj | Théorèmes limites pour les processus autorégressifs à valeurs dans D[0,1[END_REF], 2014); -administration of a drug treatment : each day produces a shock at time intervals (see Kannan and Lakshmikantham, 2002); -astronomical time series with 100000 data (see [START_REF] Preston | Discovering arbitrary event types in time series[END_REF]; -earthquake and explosion: Nason (2013); -predicting ozone [START_REF] Ignaccolo | Functional zoning for air quality[END_REF][START_REF] Dabo-Niang | Nonparametric quantile regression estimation for functional dependent data[END_REF][START_REF] Cardot | Ozone pollution forecasting using conditional mean and conditional quantiles with functional covariates[END_REF][START_REF] Damon | Estimation and simulation of autoregressive hilbertian processes with exogenous variables[END_REF]; -predicting the euro-dollar rate [START_REF] Kargin | Curve forecasting by functional autoregression[END_REF]; -finally, the mistral gust during one day or one week is one of our objective for prediction: 240000 data are at our disposal. Predicting the greatest jump should be of interest, see [START_REF] Jacq | Le mistral. Quelques aspects des connaissances actuelles[END_REF]. 1.6. In our considered framework, preliminary results were first obtained by [START_REF] Bosq | Estimating and detecting jumps. Applications to D[0,1[END_REF] and, the case of observations in continuous time also appears in [START_REF] Blanke | Exponential bounds for intensity of jumps[END_REF]. Here, we use high frequency data (HFD); this scheme appears in many situations (see [START_REF] Bollerslev | Estimation of jump tails[END_REF][START_REF] Comte | Nonparametric adaptive estimation for pure jump Lévy processes[END_REF]Aït-Sahalia and Jacod, 2009, among others). Concerning prediction with HFD, practical results will be studied later with combined predictors. In particular, we will apply the results to the mistral gusts with big data. 1.7. In Section 2, we introduce the ARMAD(1, 1) model which is connected with FDA:

X n -m -ρ(X n-1 -m) = Z n -ρ ′ (Z n-1 ), n ∈ Z (1.2)
where m is a trend and ρ(D) ⊂ C, ρ ′ (D) ⊂ C so that (X n ) and (Z n ) have the same jumps. We give several properties of (1.2) as well as examples. In the following, we study various types of jumps.

In Section 3, we consider data of the form X i ( ℓ qn ), ℓ = 0, . . . , q n , q n ≥ 1, i = 1, . . . , n; where ℓ and q n are integers and (X 1 , . . . , X n ) are D-valued realizations of (1.2). We consider the case of fixed but unknown instants of jumps t 1 , . . . , t k , where t j , denotes the j-th jump, j = 1, . . . , k and k is unknown too. In this part, each jump may occur randomly at time t j with unknown probability p j ∈]0, 1], j = 1, . . . , k, so the number of jumps is a random variable depending on i = 1, . . . , n. We propose and study detectors of jumps and next, we derive estimators of each intensity of jumps by estimating p j and plug-in the detectors.

Section 4 is devoted to the case of random instants of jumps: 0 < T 1 < T 2 < • • • < T Ki < 1 with K i a N-valued random variable. We consider the case where intensities of jumps have the same ordering in each X i . To estimate these intensities, we detect the k, k ≥ 1, first jumps by considering separately each X i . Here, as K i is random, the difficulty is to select the sample paths with at least k jumps. In this section, we also derive results for estimating the maximal jump.

In Section 5, we consider a final scheme, the completely random one where the ordering of jumps varies from each sample X i . Similarly as in the previous section, we detect the jumps with each trajectories considered separately. To estimate their intensities, their random ordering makes the problem intricate but we propose a method in the case where the number of jumps is fixed. It is based on a trick, derived from Viète's formula, that allows us to provide estimations (based on numerical approximation for a number of jumps greater than 4).

2. ARMAD(1, 1) processes 2.1. Model and properties. In order to study the jumps of the real continuous time process X = (X t , 0 ≤ t ≤ h), h > 0, we consider the space D = D([0, h]) of càdlàg real functions defined over [0, h]. The sup-norm x = sup 0≤t≤h |x(t)| entails non-separability of D. Thus, it is more convenient to use the modified Skorohod metric d • (cf Billingsley, 1999, p.125) ; with that metric, D becomes a complete separable space.

The process X being defined on the probability space (Ω, a, P ), we suppose that it is a -D measurable where D is the σ-algebra generated by d • . Concerning measurability we refer to [START_REF] Janson | Higher moments of banach space valued random variables[END_REF]. Now, if ρ is a bounded linear operator, i.e. ρ L = sup x∈D, x ≤h ρ(x) < ∞, then, it is D-D measurable. Also if there is a jump at t 0 , x → x(t 0 ) -x(t - 0 ) is a continuous linear form on (D, • ), see [START_REF] Pestman | Measurability of linear operators in the Skorokhod topology[END_REF].

We consider the ARMAD(1,1) process defined as

X n -m -ρ(X n-1 -m) = Z n -ρ ′ (Z n-1 ), n ∈ Z, (2.1)
where ρ and ρ ′ are bounded linear operators, m = E (X n ), and (Z n ) is a strong white noise i.e. the sequence (Z n ) is i.i.d., and such that E Z n 2 < ∞, E (Z n ) = 0. Note also the presence of the trend m.

In order to show existence of the ARMAD process we make the following assumption, weaker than those considered in [START_REF] Blanke | Exponential bounds for intensity of jumps[END_REF]:

Assumption 2.1 (A2.1). ∃ j 0 ≥ 1 : ρ j0 L < 1 and ∃ j 1 ≥ 1 : ρ ′j1 L < 1. Lemma 2.1. If Assumption 2.1 holds, we have X n -m = L 2 D ∞ j=0 ρ j Z n-j -ρ ′ (Z n-1-j ) , n ∈ Z, (2.2) so the process (X n -m, n ∈ Z) is stationary and (Z n , n ∈ Z) is the innovation of (X n , n ∈ Z).
Proof. To simplify the exposition, let us assume that m = 0. We may write

Y n = Z n -ρ ′ (Z n-1 ), n ∈ Z then, (Y n ) is an equidistributed sequence and Y n ≤ Z n + ρ ′ L Z n-1 , thus E Y n 2 ≤ 2E Z n 2 + 2 ρ ′ 2 L E Z n-1 2 ≤ 2(1 + ρ ′ 2 L )E Z 0 2 < ∞.
Now, we study

E j≥k+1 ρ j (Y n-j ) 2 ≤ j,j ′ ≥k+1 ρ j L ρ j ′ L E ( Y n-j Y n-j ′ ) ≤ E ( Y 0 2 )( j≥k+1 ρ j L ) 2
and A2.1 implies j≥k+1 ρ j L ----→ k→∞ 0 which gives (2.2). Finally, the condition ∃ j 1 ≥ 1 : ρ ′j1 L < 1 gives invertibility and consequently, (Z n ) is the innovation of (X n ).

We consider the following assumption:

Assumption 2.2 (A2.2). ρ(D) ⊂ C, ρ ′ (D) ⊂ C; also, m ∈ C.
From (2.1) and A2.2, one obtains for each time of jump t 0 (fixed or random)

∆ n (t 0 ) := X n (t 0 ) -X n (t - 0 ) = Z n (t 0 ) -Z n (t - 0 )
, n ∈ Z which shows that X n and Z n have the same jumps and that ∆ n (t 0 ) is i.i.d.. This assumption is reasonable since we have the following examples. Here, we may suppose that m = 0 in order to simplify the exposition.

Example 2.1. Consider the Ornstein-Uhlenbeck driven by a Levy process given by:

ξ t = t -∞
e -θ(t-s) dL(s), t ∈ R (θ > 0) [START_REF] Brockwell | Estimation for nonnegative Lévydriven Ornstein-Uhlenbeck processes[END_REF][START_REF] Cont | Financial modelling with jump processes[END_REF]) and observed over a sequence of time intervals. By using (1.1), we obtain an ARD(1) process that satisfies

X n+1 (t) = ρ θ (X n )(t) + Z n+1 (t), 0 ≤ t ≤ h, n ∈ Z,
where the linear operator ρ θ has the shape ρ θ (x)(t) = e -θt x(h), 0 ≤ t ≤ h, x ∈ D and with

Z n+1 (t) = nh+t nh e -θ(nh+t-s) dL(s), 0 ≤ t ≤ h, n ∈ Z.
Then (Z n ) is a strong white noise which may contain jumps, and since ρ θ (x) ∈ C, X n and Z n have the same jumps.

Example 2.2. Set ρ r (x)(t) = h 0 r(s, t)x(s) ds, 0 ≤ t ≤ h, x ∈ D, with |r(s, t) -r(s, t ′ )| ≤ c |t -t ′ | α , 0 < α ≤ 1, 0 ≤ t, t ′ ≤ h, c > 0, then |ρ r (x)(t ′ ) -ρ r (x)(t)| ≤ c |t -t ′ | α h 0 |x(s)| ds ------→ |t-t ′ |→0
0 since x is bounded (see Billingsley, 1999, p.122). Then, a classical example of ARMAD(1, 1) may be derived with X n -ρ r (X n-1 ) = Z n -ρ r ′ (Z n-1 ), n ∈ Z where ρ r ′ satisfies a similar condition as ρ r .

Example 2.3. Put X n+1 = ρ(X n ) + Z n+1 , n ∈ Z, where ρ j0 L < 1 for some j 0 ≥ 1. Then, it is possible to predict X n+1 by considering continuous time and jumps separately (see [START_REF] Shimizu | Threshold selection in jump-discriminant filter for discretely observed jump processes[END_REF][START_REF] Aït-Sahalia | Fisher's information for discretely sampled Lévy processes[END_REF][START_REF] Shimizu | Estimation of parameters for diffusion processes with jumps from discrete observations[END_REF]. Thus, we may suppose that ρ(D) ⊂ C.

Discrete data.

Here the data are supposed to be discrete. They take the form X i ( ℓ qn ), ℓ = 0, . . . , q n , q n ≥ 1, i = 1, . . . , n, where ℓ and q n are integers and q n → ∞ as n → ∞. Now, in all the following we set h = 1 so if (ξ t , t ∈ R) is the real measurable continuous time process such that X i (t) = ξ i-1+t , one observes ξ t at nq n + 1 discrete times 0, q -1 n , . . . , n -q -1 n , n. The instants of jumps associated with X i are denoted by T i1 , . . . , T iKi , they can be fixed or random, as well as K i , and they satisfy 0 < T i1 < • • • T iKi < 1, i = 1, . . . , n, almost surely (a.s.). Next, in order to avoid local irregularity we need the following hypothesis:

Assumption 2.3 (A2.3). For 0 < α ≤ 1, (s, t) ∈ [0, 1] 2 :
(i) For x ∈ D, the functions ρ(x), ρ ′ (x) and m are Hölderian:

|ρ(x)(t) -ρ(x)(s)| ≤ a(x) |t -s| α (a > 0), |ρ ′ (x)(t) -ρ ′ (x)(s)| ≤ b(x) |t -s| α (b > 0), |m(t) -m(s)| ≤ c m |t -s| α (c m > 0).
(ii) For i.i.d. and integrable

M i : |Z i (t) -Z i (s)| ≤ M i |t -s| α , (s, t) ∈ I iKi where I iKi = [0, T i1 [ 2 ∪ • • • ∪ [T iKi , 1[ 2 , i = 1, . . . , n. Note that Example 2.2 satisfies A2.3-(i) with a(x) = c 1 0 |x(s)
| ds and that the Ornstein-Uhlenbeck process or the fractional Brownian motion with jumps satisfies condition A2.3-(ii).

In the following, we will use repeatedly the following result since it gives a measure of proximity between increments of X and Z.

Lemma 2.2. Under the condition A2.3-(i), we have:

|X i (s) -X i (t)| -|Z i (s) -Z i (t)| ≤ (a(X i-1 ) + b(Z i-1 ) + c m I -ρ L ) |s -t| α , i = 1, . . . , n, (s, t) ∈ [0, 1] 2 .
Proof. The relation (2.1) gives

|X i (s) -X i (t)| ≤ |Z i (s) -Z i (t)| + |ρ(X i-1 )(s) -ρ(X i-1 )(t)| + |ρ ′ (Z i-1 )(s) -ρ ′ (Z i-1 )(t)| + I -ρ L |m(s) -m(t)| (2.3) next, A2.3-(i) implies |X i (s) -X i (t)| ≤ |Z i (s) -Z i (t)| + (a(X i-1 ) + b(Z i-1 ) + c m I -ρ L ) |s -t| α
, and reversing the inequality, one obtains the result.

Note that if m is constant, it disappears in (2.3) and the term c m I -ρ L is no longer relevant. One may handle this case with the choice c m = 0. The next result shows that, excluding the jump's times, X i satisfies also a Hölder type condition. It is a direct consequence of Lemma 2.2 and condition A2.3-(ii).

Corollary 2.1. Under Assumption A2.3, we have

|X i (s) -X i (t)| ≤ (a(X i-1 ) + b(Z i-1 ) + M i + c m I -ρ L ) |s -t| α , i = 1, . . . , n provided (s, t) ∈ I iK = [0, T i1 [ 2 ∪ • • • ∪ [T iK , 1[ 2 , i = 1, . . . , n.
Now, throughout the paper, we will suppose that Assumptions A2.1, A2.2 and A2.3 hold.

Fixed jumps with k unknown

3.1. Framework. In this part, we consider the model (2.1). The (Z i ) are i.i.d. functional random variables such that each Z i , i = 1, . . . , n, has at most k distinct jumps, with a fixed but unknown k ≥ 1. These jumps may occur randomly at fixed times t 1 , . . . , t k with 0 = t 0 < t 1 < . . . < t k < t k+1 = 1 such that t j+1 -t j ≥ δ 0 > 0 for all j = 1, . . . , k -1. More precisely, we set:

∆ ij = Z i (t j ) -Z i (t - j ) = X i (t j ) -X i (t - j ) = I ij Y ij , i = 1, . . . , n
where Z i (t - j ) = lim ηց0 Z i (t j -η) and (I ij , j = 1, . . . , k) are positive random variables that describe the jump amplitudes. Here, we suppose that P(I ij ≥ δ 1 ) = 1 for some positive δ 1 and, that (Y ij , j = 1, . . . , k) are independent random variables with Bernoulli distribution B(p j ),

p j ∈]0, 1], j = 1, . . . , k. Also, Y ij and I ij are independent, which means that E (∆ ij ) := E (∆ j ) = p j E (I j ) > 0, j = 1, . . . , k.
Hereafter, we present an example illustrating the considered framework.

Example 3.1 (Case k = 1). Consider n independent copies of Y 1 with B(p 1 ) distri- bution, p 1 ∈]0, 1] and (W 1 (t), W 2 (t), t ∈ [0, 1]) where W 1 and W 2 two independent C-valued processes. We set Z i (t) = W i1 (t)I [0,Ti1[ (t) + W i2 (t)I [Ti1,1[ (t), i = 1, . . . , n, t ∈ [0, 1[ with T i1 = t 1 ∈]0, 1[ if Y i1 = 1 and T i1 = 1 otherwise. In this case, intensities of jumps are given by Z i (t 1 ) -Z i (t - 1 ) = |W i1 (t 1 ) -W i2 (t 1 )| Y i1
and each sample path has at most one jump located at t 1 . Note that p 1 = 1 gives a systematic jump at t 1 . Such modeling refers to short-term perturbations that can be interpreted as impulses: for example, we may think of treatments where impulses correspond to the periodic administration of some drugs.

Finally for convenience, we suppose that (1, . . . , k). By this way, we denote by t σ(j) the jump time having the j-th intensity ∆ σ(j) , j = 1, . . . , k. Our aim is to estimate the amplitudes E (I j ), j = 1, . . . , k, on the basis of the discretely observed X 1 , . . . , X n from the model (2.1): X i ℓ qn , ℓ = 0, . . . , q n , i = 1, . . . , n, where ℓ and q n ≥ 1 are integers and lim n→∞ q n = ∞. First, we will estimate the times t jn , j = 1, . . . , k defined as:

E ∆ σ(1) > • • • > E ∆ σ(k) > 0 for some given permutation σ(1), . . . , σ(k) of
0 < ℓ jn -1 q n < t j ≤ ℓ jn q n := t jn , j = 1, . . . , k.
Here and throughout this part, we consider q n sufficiently large to have t i = t j ⇒ t i,n = t j,n and we use notation ℓ j , ℓ σ(j) for ℓ j,n , ℓ σ(j),n . Also, we set

ζ ℓ,n = 1 n n i=1 X i ( ℓ q n ) -X i ( ℓ -1 q n ) , ℓ = 1, . . . , q n
and make the assumption:

Assumption 3.1 (A3.1). (i) The distribution of ζ ℓ,n , ℓ = 1, . . . , q n is continuous. (ii) I ij ≥ δ 1 > 0 (a.s.), j = 1, . . . , k, i = 1, . . . , n
where δ 1 is fixed.

Finally for j = 1, . . . , k, or ℓ = 1, . . . , q n , we set:

∆ j,n = 1 n n i=1 X i (t j ) -X i (t - j ) and ζ (Z) ℓ,n = 1 n n i=1 Z i ℓ q n -Z i ℓ -1 q n .
We begin with a result giving the proximity between ζ

(Z) ℓj ,n and ∆ j,n . Lemma 3.1. For all j = 1, . . . , k, A2.3-(ii) implies that ζ (Z) ℓj ,n -∆ j,n ≤ 2M q -α n with M = 1 n n i=1 M i . Proof.
First, note that we have the simple inequality

|u -v| -|x -y| ≤ |u -x| + |v -y| , u, v, x, y ∈ R. (3.1) Since ∆ j,n = 1 n n i=1 X i (t j ) -X i (t - j ) = 1 n n i=1 Z i (t j ) -Z i (t - j ) , this implies ζ (Z) ℓj ,n -∆ j,n ≤ 1 n n i=1 Z i ℓ j q n -Z i ℓ j -1 q n -Z i (t j ) -Z i (t - j ) ≤ 1 n n i=1 Z i ℓ j q n -Z i (t j ) + Z i (t - j ) -Z i ℓ j -1 q n ≤ 2M q -α n from the condition A2.3-(ii)
k n → ∞ and u n → 0 such that u n q α n → ∞, for α ∈]0, 1] defined in Assumption A2.3. For example, if q n ≃ n β , β > 0, an omnibus choice for u n is u n ≃ (log n) -1 .
In order to detect the jumps, we need the following assumption. Assumption 3.2 (A3.2). Suppose that one of the following two conditions holds true:

(i) -E (a(X 1 )) < ∞, E (b(Z 1 )) < ∞, E (M 1 ) < ∞, -E ( Z 1 (t j ) -Z 1 (t - j ) 4 ) < ∞, j = 1, . . . , k, - n≥1 u -1 n q -α n < ∞. (ii) -a(X 1 ) ≤ a ∞ < ∞, E (exp(c 1 b(Z 1 ))) < ∞, E (exp(c 2 M 1 )) < ∞, (a ∞ > 0, c 1 > 0, c 2 > 0), -E (exp(c 3 Z 1 (t j ) -Z 1 (t - j ) )) < ∞, j = 1, . . . , k, (c 3 > 0).
Considering Example 2.2, A3.2(i) holds as soon as E ( 1 0 |X 1 (s)| ds) < ∞ but the condition imposed on q n implies that sample paths should be observed with high frequency, especially when α is small. Condition A3.2(ii) is more stringent since a(X 1 ) is supposed to be bounded, but in this case, the only requirement q n → ∞ is sufficient to derive exponential rates of convergence. Now, the jumps detection is carried as follows. We set ℓ 1,n = ℓ 1 = arg max ℓ=1,...,qn

ζ ℓ,n
and as ζ ℓj-1 > u n :

ℓ j = arg max ℓ=1,...,qn ℓ = ℓ1,...,ℓ = ℓj-1 ζ ℓ,n , j = 2, . . . , k n .
The number of detected jumps is then given by

k := k n = min j = 1, . . . , k n : ζ ℓj ≤ u n -1.
Remark that the unique restriction on k n is that k belongs to {1, . . . , k n } for n large enough: k n → ∞ is a sufficient condition. Hence if the above set is empty, it means that there exists at least k n jumps: in this case, from a practical point of view, one has to replace k n by k ′ n with k ′ n > k n . Finally, detectors of jumps locations are given by ( t 1,n , . . . , t k,n ) = ( t * 1,n , . . . , t * k,n ) where t * j,n is the j-th order statistic associated with ( t 1,n , . . . , t k,n ) := ( ℓ1 qn , . . . , ℓ k qn ). Note that (a.s.) uniqueness of t 1,n , . . . , t k,n is guaranteed by Assumption A3.1 and the next theorem shows that the times of jumps are detected with probability 1.

Theorem 3.1. Suppose that Assumption A3.1 holds, then the condition A3.2-(i) implies:

P k j=1 { t j,n = t j,n } = P k j=1 { t j,n = t σ(j),n } = O n -2 + O u -1 n q -α n ; (3.2)
while A3.2-(ii) gives:

P k j=1 { t j,n = t j,n } = P k j=1 { t j,n = t σ(j),n } = O exp(-c n) , c > 0. (3.3)
The same bounds hold for P( k = k) so in both cases, we obtain that a.s. for n large enough, k = k and for j = 1, . . . , k: t j,n = t σ(j),n .

Proof. We may write P(

k j=1 { ℓ j = ℓ σ(j) }) ≤ P( k j=1 { ℓ j = ℓ σ(j) }) + P( k = k). First, we have P( k = k) ≤ P( k j=1 {ζ ℓj ,n ≤ u n }) + P(ζ ℓ k+1 ,n > u n ) where P( k j=1 {ζ ℓj,n ≤ u n }) ≤ P( k j=1 {ζ ℓ σ(j) ,n ≤ u n }) + P( k j=1 { ℓ j = ℓ σ(j) }) and P(ζ ℓ k+1,n > u n ) ≤ P(ζ l k+1 ,n > u n , ∩ k j=1 { ℓ j = ℓ σ(j) }) + P( k j=1 { ℓ j = ℓ σ(j) }).
Hence,

P( k j=1 { ℓ j = ℓ σ(j) }) ≤ k j=1 P(ζ ℓ σ(j) ,n ≤ u n ) + P(ζ l k+1 ,n > u n , ∩ k j=1 { ℓ j = ℓ σ(j) }) + 3 P( k j=1 { ℓ j = ℓ σ(j) }). (3.4)
For the first term, we get from Lemma 2.2 and 3.1 that for j = 1, . . . , k

ζ ℓ σ(j) ,n ≥ ∆ σ(j),n -q -α n (2M + a X + b Z + c m I -ρ L ), (3.5)
where

a X = 1 n n i=1 a(X i ), b Z = 1 n n i=1 b(Z i ) and M = 1 n n i=1 M i . Now, we set in all the following Λ n = 2M + a X + b Z + c m I -ρ L for obtaining P(ζ ℓ σ(j) ,n ≤ u n ) ≤ P(∆ σ(j),n ≤ u n + q -α n Λ n ) ≤ P( ∆ σ(j),n -E (∆ σ(j) ) ≥ E (∆ σ(j) ) 2 ) + P(Λ n ≥ ( E (∆ σ(j) ) 2 -u n )q α n ).
These terms are controlled by the following lemma whose proof is postponed to the appendix.

Lemma 3.2. 1) Under the conditions A3.2-(i), we get for j = 1, . . . , k:

P ∆ σ(j),n -E (∆ σ(j) ) ≥ E (∆ σ(j) ) 2 = O n -2 and P Λ n > ( E (∆ σ(j) ) 2 -u n )q α n = O q -α n .
2) If the conditions A3.2-(ii) hold, then for j = 1, . . . , k:

P ∆ σ(j),n -E (∆ σ(j) ) ≥ E (∆ σ(j) ) 2 = O e -c n ,
for some c > 0 and,

P Λ n > ( E (∆ σ(j) ) 2 -u n )q α n = O e -c nq α n .
Concerning again (3.4), the term

P(ζ l k+1 > u n , ∩ k j=1 { ℓ j = ℓ σ(j) }) is controlled with ζ l k+1 ,n > u n , ∩ k j=1 { ℓ j = ℓ σ(j) } ⇒ ℓ ∈{ℓ σ(1) ,...,ℓ σ(k) } {ζ ℓ,n > u n }
and Corollary 2.1 implies that, for all ℓ ∈ {ℓ σ(1) , . . . , ℓ σ(k) }:

{ζ ℓ,n > u n } ⇒ {q -α n (a X + b Z + M + c m I -ρ L ) > u n }.
This last event does not depend on ℓ, so

P(ζ l k+1 ,n , ∩ k j=1 { ℓ j = ℓ σ(j) }) ≤ P(a X + b Z + M > u n q α n -c m I -ρ L ).
For this term, we obtain the bound O(u -1 n q -α n ) under the condition A2.3-(i) while A2.3-(ii) gives a O(e -cnunq α n ). For the last term in (3.4), observe that the property P(A∪B) = P(A c ∩B)+P(A) implies for k = 2 the relation:

P( ℓ 1 = ℓ σ(1) ∪ ℓ 2 = ℓ σ(2) ) = P( ℓ 1 = ℓ σ(1) , ℓ 2 = ℓ σ(2) ) + P( ℓ 1 = ℓ σ(1)
). Next by induction and using the convention

0 1 • • • = 0, we obtain P k j=1 { ℓ j = ℓ σ(j) } = P( ℓ 1 = ℓ σ(1) ) + k-1 j=1 P( ℓ 1 = ℓ σ(1) , . . . , ℓ j = ℓ σ(j) , ℓ j+1 = ℓ σ(j+1) ).
First part: Study of P( ℓ 1 = ℓ σ(1) ). Clearly, the relation

ζ ℓ σ(1) ,n > max ℓ=1,...,qn ℓ =ℓ σ(1) ζ ℓ,n ⇒ ℓ 1 = ℓ σ(1) gives P( ℓ 1 = ℓ σ(1) ) ≤ P ζ ℓ σ(1) ,n ≤ max ℓ=1,...,qn ℓ =ℓ σ(1)
ζ ℓ,n .

Setting j = 1 in (3.5), we obtain P(

ℓ 1 = ℓ σ(1) ) ≤ P ∆ σ(1),n ≤ max ℓ=1,...,qn ℓ =ℓ σ(1) ζ ℓ,n + q -α n Λ n . Next, we get max ℓ=1,...,qn ℓ =ℓ σ(1) ζ ℓ,n = max max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(k) ζ ℓ,n , ζ ℓ σ(2) ,n , . . . , ζ ℓ σ(k) ,n ≤ max q -α n (a X + b Z + M + c m I -ρ L ), ζ ℓ σ(2) ,n , . . . , ζ ℓ σ(k) ,n .
(3.6) On the other hand, from Lemma 2.2 and Lemma 3.1, we get for all j ≥ 2,

ζ ℓ σ(j) ,n ≤ ∆ σ(j),n + q -α n (2M + a X + b Z + c m I -ρ L ). (3.7)
We may deduce that max ℓ=1,...,qn ℓ =ℓ σ(1)

ζ ℓ,n ≤ max j=2,...,k
∆ σ(j),n + q -α n Λ n and, finally, we obtain that

P( ℓ 1 = ℓ σ(1) ) ≤ P(∆ σ(1),n ≤ max j=2,...,k ∆ σ(j),n + 2q -α n Λ n ) (3.8)
and

P( ℓ 1 = ℓ σ(1) ) ≤ P(∆ σ(1),n ≤ ∆ σ(2),n + 2q -α n Λ n ) + P(∆ σ(2),n < max j=3,...,k ∆ σ(j),n ).
Note that (3.8) reduces to P(∆ σ(1),n ≤ 2q -α n Λ n ) if k = 1: this particular case will be handled in the second part of the proof. Here, since ∆ σ(2),n < max j=3,...,k

∆ σ(j),n ⇔ ∃ j = 3, . . . , k, ∆ σ(j),n > ∆ σ(2),n , we get P( ℓ 1 = ℓ σ(1) ) ≤ P ∆ σ(2),n -∆ σ(1),n -E (∆ σ(2) -∆ σ(1) ) ≥ E (∆ σ(1) -∆ σ(2) ) -2q -α n Λ n + k j=3 P ∆ σ(j),n -∆ σ(2),n -E (∆ σ(j) -∆ σ(2) ) ≥ E (∆ σ(2) -∆ σ(j) ) . (3.9) By considering the event {E (∆ σ(1) -∆ σ(2) ) -2q -α n Λ n ≥ 1 2 E (∆ σ(1) -∆ σ(2)
)}, we may bound the first term of (3.9) by

P ∆ σ(2),n -∆ σ(1),n -E (∆ σ(2) -∆ σ(1) ) ≥ E (∆ σ(1) -∆ σ(2) ) 2 + P Λ n > E (∆ σ(1) -∆ σ(2) ) 4
q α n . These probabilities are controlled by the following lemma whose proof is postponed to the Appendix. Lemma 3.3. For all j = 2, . . . , k, j ′ = 1, . . . , j -1 and η > 0, the following bounds hold.

1) If the conditions given in A3.2-(i) are fulfilled, P Λ n > ηq α n = O q -α n and

P ∆ σ(j),n -∆ σ(j ′ ),n -E (∆ σ(j) -∆ σ(j ′ ) ) ≥ η = O n -2 .
2) If the conditions A3.2-(ii) hold, P Λ n > ηq α n = O e -c nq α n and for some c > 0:

P ∆ σ(j),n -∆ σ(j ′ ),n -E (∆ σ(j) -∆ σ(j ′ ) ) ≥ η = O e -c n .
Finally, the last term of (3.9) (which exists only for k ≥ 3) is also derived from Lemma 3.3. Consequently, the control of P( ℓ 1 = ℓ σ(1) ) is achieved by collecting all the previous results and Borel-Cantelli's lemma implies that a.s. for n large enough, ℓ 1 = ℓ σ(1) .

Second part: Study of

k-1 j=1 P(∩ j m=1 { ℓ m = ℓ σ(m) } ∩ { ℓ j+1 = ℓ σ(j+1) }) for k ≥ 2.
For this term, we have

∩ j m=1 { ℓ m = ℓ σ(m) } ∩ { ℓ j+1 = ℓ σ(j+1) } = ∩ j m=1 { ℓ m = ℓ σ(m) } ∩ {arg max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(j) ζ ℓ,n = ℓ σ(j+1) }. As {ζ ℓ σ(j+1) ,n > max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(j+1) ζ ℓ,n } ⇒ {arg max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(j) ζ ℓ,n = ℓ σ(j+1) }, we
deduce that the probability of interest is bounded by

P ζ ℓ σ(j+1) ,n ≤ max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(j+1) ζ ℓ,n .
Then, using the convention

0 1 • • • = 0, it is sufficient to control the terms k-2 j=1 P ζ ℓ σ(j+1) ,n ≤ max max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(k) ζ ℓ,n , ζ ℓ σ(j+2) ,n , . . . , ζ ℓ σ(k) ,n + P ζ ℓ σ(k) ,n ≤ max ℓ=1,...,qn ℓ =ℓ σ(1) ,...,ℓ =ℓ σ(k) ζ ℓ,n .
Using the bounds established in (3.5)-(3.7), we arrive at

k-2 j=1 P ∆ σ(j+1),n ≤ max m=j+2,...,k ∆ σ(m),n + 2q -α n Λ n + P E ∆ σ(k) -∆ σ(k),n ≥ E ∆ σ(k) -2q -α n Λ n where again Λ n = 2M + a X + b Z + c m I -ρ L .
The study of the first term is analogous to that performed for the term given in (3.8). Details are left to the reader. The second one is handled similarly to (3.9) for obtaining:

P ∆ σ(k),n -E ∆ σ(k) ≥ E ∆ σ(k) 2 + P Λ n ≥ E ∆ σ(k) 4
q α n and the upper bounds are the same as those established in Lemma 3.3. Collecting all the results, Borel Cantelli's lemma applied to (3.4) implies that n P( k = k) < ∞ and n P k j=1 { ℓ j = ℓ σ(j) } < ∞ leading to the final result.

3.3. Estimation of intensity. Since a.s. for n large enough, k = k and consecutive times of jumps are detected with ( ℓ 1 , . . . , ℓ k ) = ( ℓ * 1 , . . . , ℓ * k ) the associated order statistic, we may evaluate their corresponding intensities E (I j ), j = 1, . . . , k. We start by estimating E (∆ j ) with

∆ j = 1 n n i=1 X i ℓ j q n -X i ℓ j -1 q n , j = 1, . . . , k.
Since E (∆ j ) = p j E (I j ), estimators of E (I j ) are given by

I j = ∆ j p j where p j = 1 n n i=1 I |Xi( ℓ j qn )-Xi( ℓ j -1 qn )|>un , j = 1, . . . , k
with the same u n as in Section 3.2, satisfying again the condition: u n → 0 such that u n q α n → ∞. Note that p j a.s.

----→ n→∞ p j so, a.s. for n large enough, the denominator is not zero.

For the almost sure behavior, we study the quantity

I j -E (I j ) = ( ∆ j -E (∆ j )) -( p j -p j )E (I j ) p j , j = 1, . . . , k,
and for ε > 0, we get

P( k j=1 I j -E (I j ) ≥ ε) ≤ P( k = k) + k j=1 P( I j -E (I j ) ≥ ε) ≤ P( k = k) + k j=1 P( ∆ j -E (∆ j ) ≥ ε p j 2 ) + P(| p j -p j | ≥ ε p j 2E (I j ) )
and for all η ∈]0, p j [, we have

≤ P( k = k) + k j=1 P( ∆ j -E (∆ j ) ≥ ε(p j -η) 2 ) + P(| p j -p j | ≥ ε(p j -η) 2E (I j ) ) + 2 P(| p j -p j | ≥ η) (3.10)
where the latter term doest not depend on ε. Then we may derive the following result whose proof is postponed to the Appendix.

Theorem 3.2. Under the Assumption A3.1, we obtain 1) if the condition A3.2-(i) holds, and u n = (log n) -1 , q n = n β with β > 5 4α , then almost surely for n large enough

I j -E (I j ) = O (log n) c n 1 4 , c > 1 4 , j = 1, . . . , k;
2) if the condition A3.2-(ii) holds, and u n q α n → ∞, then almost surely for n large enough

I j -E (I j ) = O log n n , j = 1, . . . , k.
We conclude that, under the mild conditions A3.2-(i), one needs to observe each sample path with high frequency to estimate the intensities of jumps with some given accuracy. Recall that α is linked with regularity of the process between two jumps. Looking at the condition β > 5 4α , it appears, as expected, that more α is small, more the estimation will be difficult without a high sampling rate. Under A3.2-(ii) with the boundedness of a(X 1 ), we are close to the classical root-n rate of convergence. Finally by examining the proof of Theorem 3.2, it appears that the strong consistency of I j holds in both cases as soon as q n → ∞.

Random jumps

4.1. Detection of jumps. Now, we suppose that Z i , i = 1, . . . , n, has K i jumps at random instants, with K i a nonnegative integer-valued random variable and 0

< T i,1 < • • • < T i,K i < 1 almost surely if K i ≥ 1. Here the sequence ( Z i (T ij ) -Z i (T - ij ) , K i ) i = 1, . . . , n , with Z(T - i,j ) = lim ηց0 Z(T i,j -η), is i.i.d.
. We set p k = P(K i = k) for k ≥ 0, p 0 = 1, and also, we suppose that

K i is independent from Z i (T ij ) -Z i (T - ij ) , j = 1, . . . , K i , i = 1, . . . , n. Assumptions A2.1-A2.2 lead to Z i (T ij ) -Z i (T - ij ) = X i (T ij ) -X i (T - ij ) .
Recall that the trajectories satisfy a Hölder condition between two consecutive jumps. The main difference with the previous section is that times of jumps differ from one sample path to the other. By this way, we have to consider separately the X i 's for their detection. Finally, we associate to each T i,j an integrable intensity of jump:

∆ ij = X i (T i,j ) -X i (T - i,j ) = Z i (T i,j ) -Z i (T - i,j ) , j = 1, . . . , K i ; i = 1, . . . , n with P(∆ 1j > δ 1 ) = 1 for some δ 1 > 0. Example 4.1. Let 0 = T i0 < T i1 < • • • be a strictly increasing sequence of random variables (a.s.). Let us set K i = ∞ j=1 I Tij ≤1 and Z i (t) = k+1 j=1 Y i,j-1 I [Ti,j-1,Tij [ (t) if K i = k with T i,k+1 = 1, 0 ≤ t ≤ 1, where Y ij is A -B R
measurable and for each j = 1, . . . , k, Y 1j , . . . , Y nj are i.i.d.. Note that an example of such a model is the compound Poisson process. Now, for ℓ = 1, . . . , q n , j = 1, . . . , K i and i = 1, . . . , n, we set:

ζ iℓn = X i ℓ q n -X i ℓ -1 q n , ζ (Z) iℓn = Z i ℓ q n -Z i ℓ -1
q n and we consider the integer-valued variables L ijn defined as:

L ijn -1 q n < T i,j ≤ L ijn q n , j = 1, . . . , K i , i = 1, . . . , n. (4.1)
We associate them with the increments

ζ iLj n = X i Lijn qn -X i Lijn-1 qn and ζ (Z) iLj n = Z i Lijn qn -Z i Lijn-1 qn
. Thus, these variables correspond to the increments including a jump. To detect these jumps, the following conditions will be useful instead of Assumption A3.1 and A3.2. Assumption 4.1 (A4.1).

(i) W ij = T i,j -T i,j-1 ≥ δ 0 , j = 1, . . . , K i + 1, where T i,0 = 0, T i,Ki+1 = 1 and δ 0 is a positive constant. (ii) ∆ ij ≥ δ 1 > 0 (a.s.), j = 1, . . . , K i , i = 1, . . . , n where δ 1 is fixed.

Assumption A4.1 means that W ij and ∆ ij are not too small. Here and throughout this section, we take n large enough (namely such that 1 qn < δ 1 ) to make sure that all intervals [ ℓ-1 qn , ℓ qn ], ℓ = 1, . . . , q n include at most one jump. The first condition can be relaxed as shown by the following remark.

Remark 4.1. The condition A4.1-(i) excludes in particular gamma-distributed interarrival times. By adding the condition n≥1 nq -1 n < ∞, observe that all subsequent results of this part hold true as soon as

P(T i,j+1 -T i,j < q -1 n | K i = k) ≤ ψ(k)q -1 n with ψ such that E (K 1 Ψ(K 1 )) < ∞.
A compound Poisson process satisfies this condition since we have

P(T i,j+1 -T i,j < q -1 n | K i = k) = 1 -(1 -q -1 n ) k ≤ kq -1 n and E (K 2 1 ) < ∞.
Assumption 4.2 (A4.2). Suppose that for some p ≥ 1:

(i) E a(X 1 ) p < ∞, E b(Z 1 ) p < ∞, E M 1 p < ∞, (ii) n≥1 nq -α p n u -p n < ∞.
The condition A4.2-(ii) implies that more α is small (more the sample paths are irregular), more p should be chosen large enough. Now, to detect the jumps we consider the random set L in defined by L in = L ijn , j = 1, . . . , K i , K i ≥ 1 , i = 1, . . . , n and we predict this set with

L in = ℓ ∈ {1, . . . , q n } : X i ( ℓ q n ) -X i ( ℓ -1 q n ) > u n
still with u n → 0 such that u n q α n → ∞ for α ∈]0, 1] defined in Assumption A2.3. Again an omnibus choice is u n = (log n) -1 for q n = n β , β > 0. Moreover we denote by K i the cardinal of the set L in and { L i1 , . . . , L i Ki } its elements. We begin with a result enlightening the fact that for each sample path and n large enough, one may identify the K i jumps with probability 1.

Theorem 4.1. If Assumptions A4.1 and A4.2-(i) hold, P n i=1 L in ≡ L in = O(nq -αp n u -p n ).
Proof. We have

L in ≡ L in ⇔ (a) ∀ j = 1, . . . , K i , L ijn ∈ L in (K i ≥ 1) (b) ∀ ℓ ∈ L in , ℓ ∈ L in .
We may deduce that

n i=1 L in ≡ L in = n i=1 Ki j=1 {L ijn ∈ L in } ∪ qn ℓ=1,ℓ ∈Lin {ℓ ∈ L in } . Moreover Ki j=1 {L ijn ∈ L in } ⇔ Ki j=1 X i ( Lijn qn ) -X i ( Lijn-1 qn ) ≤ u n and qn ℓ=1 ℓ ∈Lin {ℓ ∈ L in } ⇔ qn ℓ=1 ℓ ∈Lin { X i ( ℓ q n ) -X i ( ℓ -1 q n ) > u n } . Hence P( n i=1 { L in ≡ L in }) ≤ n i=1 p i1n + p i2n with p i1n := P( Ki j=1 { X i ( L ijn q n ) -X i ( L ijn -1 q n ) ≤ u n })
and

p i2n := P( qn ℓ=1,ℓ ∈Lin { X i ( ℓ q n ) -X i ( ℓ -1 q n ) > u n }).
Let us begin by p i1n : from Lemma 2.2, we have:

-(a(X i-1 ) + b(Z i-1 ) + c m I -ρ L )q -α n + Z i ( L ijn q n ) -Z i ( L ijn -1 q n ) ≤ X i ( L ijn q n ) -X i ( L ijn -1 q n ) ≤ (a(X i-1 ) + b(Z i-1 ) + c m I -ρ L )q -α n + Z i ( L ijn q n ) -Z i ( L ijn -1 q n ) .
Moreover similarly to Lemma 3.1 we may deduce from (3.1) that

Z i ( L ijn q n ) -Z i ( L ijn -1 q n ) -Z i (T ij ) -Z i (T - ij ) ≤ 2M i q -α n . (4.2) Setting Λ i = a(X i-1 ) + b(Z i-1 ) + 2M i + c m I -ρ L , we get -Λ i q -α n + Z i (T ij ) -Z i (T - ij ) ≤ X i ( L ijn q n ) -X i ( L ijn -1 q n ) ≤ Λ i q -α n + Z i (T ij ) -Z i (T - ij ) (4.3)
and p i1n is bounded as follows:

p i1n ≤ P( Ki j=1 Z i (T ij ) -Z i (T - ij ) ≤ u n + Λ i q -α n ) ≤ P( Ki j=1 { Z i (T ij ) -Z i (T - ij ) ≤ 2u n }) + P(Λ i > q α n u n ) ≤ ∞ k=0 P( k j=1 { Z i (T ij ) -Z i (T - ij ) ≤ 2u n } | K i = k)P(K i = k) + P(Λ i > q α n u n ) ≤ ∞ k=1 k j=1 P(∆ ij ≤ 2u n )P(K i = k) + P(Λ i > q α n u n ).
because K i is independent from ∆ ij . Next Assumption A4.1-(ii) implies the nullity of the first term for n large enough (namely such that 2u n ≤ δ 1 ) and the second term is controlled by Markov's inequality and Assumption A4.2-(i). Hence, we arrive at p i1n = O(q -αp n u -p n ) uniformly in i. Now, we turn to p i2n . From Corollary 2.1, we know that

X i ( ℓ q n ) -X i ( ℓ -1 q n ) ≤ (a(X i-1 ) + b(Z i-1 ) + M i + c m I -ρ L )q -α n so qn ℓ=1,ℓ ∈Lin X i ( ℓ q n ) -X i ( ℓ -1 q n ) > u n ⇒ a(X i-1 ) + b(Z i-1 ) + M i + c m I -ρ L > q α n u n
and,

p i2n ≤ P(a(X i-1 ) > v n ) + P(b(Z i-1 ) > v n ) + P(M i > v n ) with v n = unq α n -cm I-ρ L 3
and Markov's inequality gives that p i2n has a similar order as p i1n . 4.2. Estimation of intensity. As (∆ ij , K i ), i = 1, . . . , n is supposed to be i.i.d., we have

E (∆ ij ) = E X i (T ij ) -X i (T - ij ) ≡ E (∆ 1j
), j = 1, . . . , K i , i = 1, . . . , n. So, the ordering of jumps' intensities is the same for each sample path; but contrary to the deterministic case, two distinct jumps may have the same intensity. Again Assumptions A2.1 and A2.2 guarantee that for each j = 1, . . . , K i ,

X i (T ij ) -X i (T - ij ) = Z i (T ij ) -Z i (T - ij )
are independent variables. For some fixed k ≥ 1, it is possible to construct an estimator of the k-first jumps E (∆ 1 ), . . . , E (∆ k ) by selecting the X i 's having at least k jumps. To this end, we set for j = 1, . . . , k:

∆ j := ∆ jn =        n i=1 X i Lij qn -X i Lij -1 qn I { Ki≥j} n i=1 I { Ki≥j} , if n i=1 I { Ki≥j} > 0, 0, if n i=1 I { Ki≥j} = 0, still with K i = L in and L in = L i1 , . . . , L i Ki , i = 1, . . . , n.
The strong consistency and rates of convergence are given in the following theorem.

Theorem 4.2. Suppose that Assumptions A4.1 and A4.2 (with p = 1) are fulfilled, and that for j = 1, . . . , k E (exp(c 0 ∆ 1j )) < ∞ with c 0 > 0. We have for all ε > 0;

P ∆ j -E (∆ 1j ) ≥ ε = O(nq -α n u -1 n ) + O(exp(-c 1 nε 2 )) + O( n log n q α n ε ), c 1 > 0.
Proof.

We have to study

P ∆ j -E (∆ 1j ) ≥ ε , j = 1, . . . , k, k ≥ 1, ε > 0.
First, this term is equal to

P ∆ j -E (∆ 1j ) ≥ ε, n i=1 L i ≡ L in + P ∆ j -E (∆ 1j ) ≥ ε, n i=1 L i ≡ L in so it may be bounded with P( n i=1 L i ≡ L in ) + P( ∆ j -E (∆ 1j ) ≥ ε ) where we have set ∆ j = n i=1 X i ( Lij qn ) -X i ( Lij -1 qn ) I {Ki≥j} n i=1 I {Ki≥j} I n i=1 I {K i ≥j}>0
using the convention 0 0 = 0. The first term is controlled with Theorem 4.1 and gives a O(nq -α n u -1 n ). Next from (4.3) and after some derivations, we may write

P( ∆ j -E (∆ 1j ) ≥ ε) ≤ p 1n + p 2n with p 1n := P( n i=1 ∆ ij I {Ki≥j} n i=1 I {Ki≥j} I n i=1 I {K i ≥j}>0 -E (∆ 1j ) ≥ ε 2 ) p 2n := P( n i=1 Λ i I {Ki≥j} n i=1 I {Ki≥j} I n i=1 I {K i ≥j}>0 ≥ q α n ε 2 ).
Concerning the first term p 1n , we have

p 1n = n m=0 P n i=1 ∆ i,j I {Ki≥j} m I m>0 -E (∆ 1j ) ≥ ε 2 | n i=1 I {Ki≥j} = m × P( n i=1 I {Ki≥j} = m).
As n i=1 I {Ki≥j} ∼ B(n, i≥j p i ) and, since n i=1 I {Ki≥j} = m is equivalent to have exactly m indicators equal to 1, the i.i.d assumption on the ∆ i,j 's and independence from K i give

p 1n = I {ε≤2E (∆1j)} P( n i=1 I {Ki≥j} = 0 + n m=1 P( m i=1 ∆ i,j m -E (∆ 1j ) ≥ ε 2 ) × P( n i=1 I {Ki≥j} = m).
Now, one may use Bernstein's inequality, stated as in e.g. Bosq and Blanke (2007, p.297), to obtain:

p 1n ≤ (1 - i≥j p i ) n + 2 n m=1 n m (1 - i≥j p i ) n-m ( i≥j p i ) m exp - mε 2 8σ 2 j + 4H j ε with σ 2 j = Var (∆ 1j
) and H j a constant linked to the central moments of ∆ 1j . The last expression is bounded by 2 1 -i≥j p i + i≥j p i exp -

ε 2 8σ 2 j +4Hj ε n . Since
ln(1 -a) ≤ -a for 0 < a < 1 and 1 -e -a ≥ a -a 2 2 for all a ≥ 0, we successively obtain for all j such that i≥j p i > 0:

p 1n ≤ 2 exp -n i≥j p i 1 -exp(- ε 2 8σ 2 j + 4H j ε ) ≤ 2 exp - n i≥j p i ε 2 8σ 2 j + 4H j ε 1 - ε 2 16σ 2 j + 8H j ε .
Next, there exists 0 < c 2 < 1 such that p 1n ≤ 2 exp -c 2 n( i≥j p i )

ε 2 8σ 2 j +4Hj ε .
Finally, for the term p 2n we may write:

p 2n = n m=1 P( n i=1 Λ i I {Ki≥j} m ≥ q α n ε 2 , n i=1 I {Ki≥j} = m) ≤ n m=1 P( n i=1 Λ i I {Ki≥j} ≥ m 2 q α n ε).
We conclude with Markov's inequality and the condition A4.2-(i), p = 1, to get the bound O( n log n q α n ε ). Remark 4.2. We may observe that the choices

u n = (log n) -1 , q n = n β , ε = ε 0 n -1 2 (log n) γ (ε 0 > 0), with γ > 2, β ≥ 5 2α entail n P ∆ j -E (∆ 1j ) ≥ ε 0 n -1 2 (log n) γ < ∞ . So
P( ∆ max -E (∆ 1kmax ) ≥ ε) ≤ kn j=1 P( ∆ j -E (∆ 1j ) ≥ ε).
Now, for K i with a finite support {0, . . . , k 0 } and unknown k 0 ≥ 1, we clearly have ∆ max = max j=1,...,k0

∆ j almost surely for n large enough (as a consequence of K i = K i giving in turn that ∆ j = 0 for n large enough and j ≥ k 0 + 1). Also, remark that max j=1,...,kn

E (∆ 1j ) = max j=1,...,k0 E (∆ 1j
) and E (∆ 1j ) = 0 for j ≥ k 0 + 1. Hence the summation ranges over [[0, k 0 ]] and one may obtain a similar rate of convergence as in Remark 4.2 for the estimation of the maximal jump. If K i is a N-valued random variable, we can also derive a rate of convergence with the same methodology as in [START_REF] Blanke | Exponential bounds for intensity of jumps[END_REF] and with sequences k n increasing slowly to infinity. Finally, it can also be shown that k max = arg max j=1,...,kn ∆ j is a consistent estimator of k max .

5. The completely random case 5.1. The considered framework. In this part, for a fixed k ≥ 1, we denote by (∆ σ(1) , . . . , ∆ σ(k) ), k independent intensities of jumps which are ordered in decreasing average:

E ∆ σ(1) > • • • > E ∆ σ(k)
. We associate them to k independent continuous variables (T σ(1) , . . . , T σ(k) ): by this way, T σ(j) corresponds to the jump with highest j-th average intensity. Next, with the ordered statistics (T 1 , . . . , T k ) = (T * σ(1) , . . . , T * σ(k) ),

T * σ(1) < • • • < T * σ(k)
we consider a sample path Z with jumps at times (T 1 , . . . , T k ). Then, we work with n i.i.d copies of Z, say Z 1 , . . . , Z n . Here, the key difference with the random case is that intensities of jumps have not the same order from one sample path to the other and the difficulty is to estimate them. The latter construction is resumed with the following hypothesis.

Assumption 5.1 (A5.1). For each i = 1, . . . , n, there exists a permutation denoted by σ i (1), . . . , σ i (k) of (1, . . . , k) such that E ∆ iσi(j) = E ∆ σ(j) with E ∆ σ(1) > • • • > E ∆ σ(k) . Moreover (∆ iσi (j) , j = 1, . . . , k, i = 1, . . . , n) is a collection of independent random variables and, for each j = 1, . . . , k, the (∆ iσi(j) , i = 1, . . . , n) are identically distributed.

We make use of the L ijn 's defined in equation (4.1), linked with the arrival times of jumps (in chronological order) and, we consider their independent counterparts L iσi(j)n with

L iσ i (j)n -1 qn < T iσi(j) ≤ L iσ i (j)n qn
, i = 1, . . . , n, j = 1, . . . , k (associated with jumps ordered by intensities). Now, we suppose that Assumption 5.2 (A5.2).

(i) (T iσi(j) , i = 1, . . . , n, j = 1, . . . , k) are globally independent with respective bounded densities f 1 , . . . ,

f k on [0,1]. (ii) n≥1 nq -1 n < ∞. (iii) ∆ ij ≥ δ 1 , j = 1, . . . , k, i = 1, . . . , n where δ 1 is a positive constant.
The next lemma establishes that with probability one, two consecutive instants are not in the same interval.

Lemma 5.1. If the conditions (i)-(ii) of Assumption A5.2 hold, for all i = 1, . . . , n, the (T ij , j = 1, . . . , k) do not belong to the same interval a.s. for n large enough:

P n i=1 k j=1 {T i,j+1 -T ij ≤ 1 qn } = O(nq -1 n ).
Proof. Note that

n i=1 k j=1 T i,j+1 -T ij ≤ 1 qn ⇒ n i=1 k j,j ′ =1 j ′ =j qn ℓ=1 {T iσi(j) ∈ [ ℓ-1 qn , ℓ qn ] ∩ T iσi(j ′ ) ∈ [ ℓ-1 qn , ℓ qn ]} .
Using independence and boundedness of the densities of T iσ(j) 's, we get that

n i=1 j =j ′ qn ℓ=1 P(T iσi(j) ∈ [ ℓ-1 qn , ℓ qn ] ∩ T iσi(j ′ ) ∈ [ ℓ-1 qn , ℓ qn ]) = O(nq -1
n ). 5.2. Detection of jumps. We begin with a result enlightening the fact that for each sample path, one may identify the k jumps with probability 1 for n large enough. Again in this part, the set L in is defined by L in = L ijn , j = 1, . . . , k , i = 1, . . . , n, ζ iℓn = X i ( ℓ qn ) -X i ( ℓ-1 qn ) and we note ζ iLj n := ζ i,Lijn,n . Theorem 5.1. Suppose that Assumptions A4.2, A5.1 and A5.2 are fulfilled, then a.s. for n large enough, we get that ζ iℓn < ζ iLj n , j = 1, . . . , k, i = 1, . . . , n, ℓ = 1, . . . , q n with ℓ ∈ L in . More precisely,

P n i=1 qn ℓ=1 ℓ ∈Lin k j=1 ζ iℓn ≥ ζ iLj n = O(nq -αp n ) + O(nq -1 n ).
Proof.

ζ iℓn ≥ ζ iLj n , n i=1 k j=1 {T i,j+1 -T ij > 1 q n } + P n i=1 k j=1 {T i,j+1 -T ij ≤ 1 q n } .
Next, from Lemma 2.2, we may write for all ℓ = 1, . . . , q n , and i = 1, . . . , n: max ℓ=1,...,qn ℓ ∈Lin

ζ iℓn ≥ ζ iLj n ⇒ max ℓ=1,...,qn ℓ ∈Lin ζ (Z) iℓn + q -α n (a(X i-1 ) + b(Z i-1 ) + c m I -ρ L ) ≥ ζ iLj n . As for ℓ ∈ L in ,
ζ iℓn ≥ ζ iLj n ⇒ q -α n (M i + a(X i-1 ) + b(Z i-1 ) + c m I -ρ L ) ≥ ζ iLj n .
(5.1) Next we may use (4.2) (since

n i=1 k j=1 {T i,j+1 -T ij > 1
qn } implies that two consecutive jumps cannot belong to the same interval) and deduce with Lemma 2.2 that

ζ iLj n ≥ ∆ ij -q -α n (2M i + a(X i-1 ) + b(Z i-1 ) + c m I -ρ L ).
Hence, (5.1) may be rewritten as max ℓ=1,...,qn ℓ ∈Lin

ζ iℓn ≥ ζ iLj n ⇒ q -α n (3M i +2a(X i-1 )+2b(Z i-1 )+2c m I -ρ L ) ≥ ∆ ij .
Finally the condition A5.2-(iii) gives that P max ℓ=1,...,qn ℓ ∈Lin

ζ iℓn ≥ ζ iLj n is bounded with P(M i ≥ δ 1 q α n -6c m I -ρ L 9 ) + P(a(X i-1 ) ≥ δ 1 q α n -2c m I -ρ L 6 ) + P(b(Z i-1 ) ≥ δ 1 q α n -2c m I -ρ L 6
).

The result follows with Markov's inequality, the condition A4.2-(i) and the conclusion is a straightforward consequence of Borel Cantelli's lemma with A4.2-(ii).

Remark 5.1. Theorem 5.1 implies that almost surely for n large enough, ζ iℓn < ζ iLj n for all ℓ ∈ L in , i = 1, . . . , n, j = 1, . . . , k. Hence for each sample path, the jumps are almost surely identified and we have at our disposal n sets of k values: { L i1 , . . . , L ik }. Here, note that the L ij are not ordered either with respect to jumps intensities or arrival times. By considering the associated order statistics: ( L i1 , . . . , L ik ) := ( L * i1 , . . . , L * ik ), the set L in := { L i1 , . . . , L ik } represents the arrival times of jumps and one gets L in ≡ L in , i = 1, . . . , n a.s. for n large enough. 5.3. Estimation of the jumps' intensities. Since we may identify a.s. for n large enough the k jumps of each X i , we are in position to estimate the intensities E (∆ σ(j) ), j = 1, . . . , k. We begin with the estimation of coefficients a 0 , . . . , a k , a k = 1, of the polynomial of degree k with the distinct roots E (∆ σ(j) ):

k j=1 (x -E ∆ σ(j) ) = k j=0 a k-j x k-j = 0.
Using Viète's formula and independence of the jumps, we get for j = 1, . . . , k:

a k-j = (-1) j 1≤ℓ1<•••<ℓj≤k E (∆ σ(ℓ1) ) • • • E (∆ σ(ℓj ) ) = (-1) j 1≤ℓ1<•••<ℓj ≤k E (∆ σ(ℓ1) • • • ∆ σ(ℓj ) ).
Here, the key point is that we have to consider the sum of k jumps, the sum of their product in pairs, ..., and finally their products. All these sums are exhaustive, hence we observe that we may use the jumps estimated by chronological order to estimate each term. The next example illustrates this fact for k = 2 and k = 3.

Example 5.1.

-For k = 2, we get

a 0 = E (∆ σ(1) ∆ σ(2) ) = E (∆ 1 ∆ 2 ), a 1 = E (∆ σ(1) + ∆ σ(2) ) = E (∆ 1 + ∆ 2 ), a 2 = 1; -for k = 3, a 0 = E (∆ σ(1) ∆ σ(2) ∆ σ(3) ) = E (∆ 1 ∆ 2 ∆ 3 ), a 1 = E (∆ σ(1) ∆ σ(2) + ∆ σ(1) ∆ σ(3) + ∆ σ(2) ∆ σ(3) ) = E (∆ 1 ∆ 2 + ∆ 1 ∆ 3 + ∆ 2 ∆ 3 ), a 2 = E (∆ σ(1) + ∆ σ(2) + ∆ σ(3) ) = E (∆ 1 + ∆ 2 + ∆ 3 ), a 3 = 1.
Hence, we compute the k estimators of a k-j , j = 1, . . . , k by setting a k-j equal to

1≤ℓ1<•••<ℓj ≤k (-1) j n n i=1 X i ( L iℓ1n q n )-X i ( L iℓ1n -1 q n ) • • • X i ( L iℓj n q n )-X i ( L iℓj n -1 q n ) .
To study their behavior, we use the Remark 5.1 and the property that summations are exhaustive to obtain below the strong consistency of these estimators as well as their rates of convergence.

5.3.1.

Convergence of the a k-j , j = 1, . . . , k. For a k-1 defined by

a k-1 = - 1 n n i=1 k j=1 X i ( L ijn q n ) -X i ( L ijn -1 q n ) and a k-1 = -E ( k j=1 ∆ σ(j) ) = -E ( k j=1 ∆ j )
, we obtain the following result proved in the Appendix.

Proposition 5.1. Suppose that Assumptions A4.2, A5.1, and A5.2 are fulfilled, then 1) a k-1 a.s.

----→ n→∞ a k-1 if either a(•) is bounded or n q -α n < ∞;

Then, the left probability of (A.3) is bounded with ).

P( 1 n n i=1 I {Yij =1} -p j > ε 1 3E (I 1j
The first term is a O(exp(-2n

ε 2 1 9E (I1j ) 2
)) by Hoeffding's inequality. For the others, we have for Λ i = a(X i-1 ) + b(Z i-1 ) + 2M i + c m I -ρ L :

Z i (t j ) -Z i (t - j ) -Λ i q -α n ≤ X i ( ℓ j q n ) -X i ( ℓ j -1 q n ) ≤ Z i (t j ) -Z i (t - j ) + Λ i q -α
n so we get the two implications:

|X i ( ℓ j q n ) -X i ( ℓ j -1 q n )| > u n , Z i (t j ) -Z i (t - j ) = 0 ⇒ Λ i > u n q α n |X i ( ℓ j q n ) -X i ( ℓ j -1 q n )| ≤ u n , Z i (t j ) -Z i (t - j ) = I ij ⇒ Λ i ≥ q α n (I ij -u n ) ⇒ Λ i ≥ q α n (δ 1 -u n )
since P(I ij > δ 1 ) = 1. Under the condition A3.2-(i), we arrive at a bound of order O(u -1 n q -α n ε -1 1 )) + O(exp(-cnε 2 1 )) + O(u -1 n q -α n ) + O(n -2 ) for the term given in (A.3). Finally, collecting all the results, the predominant bounds are of order O(u -1 n q -α n ε -1 1 ) + O(n -2 ε -4 1 ). Next setting ε 1 = (log n) c n -1 4 , c > 1 4 , and q n = n β with β > 5 4α , u n = (log n) -1 , we may apply Borel-Cantelli's lemma to derive the claimed result. If the condition A3.2-(ii) is fulfilled, the predominant bound is now transformed in O(exp(-cnε 2 1 ), so we may derive the rate of convergence with the choice ε 1 = ε 0 log n n for a sufficiently large enough ε 0 and all q n → ∞ (since nε 2 1 = o(nq α n ε 1 )).

Proof of Proposition 5.1. 1) To get the strong consistency, we notice that a.s. for n large enough as all possible summations in j are considered. From Lemma 2.2 and Lemma 3.1 (whose proof is exactly the same for random instants of jumps), we obtain the same bound for each j = 1, . . . , k: .4) with again Λ n = 2M + a X + b Z + c m I -ρ L . We conclude with the law of large numbers (applying Markov's inequality to control a X in the case where a(•) is not bounded.

ζ Lj n -E ∆ j -∆ j -E ∆ j ≤ 2Λ n q -α n (A
2) We have

P | a k-1 -a k-1 | ≥ ε = P | a k-1 -a k-1 | ≥ ε, n i=1 L in ≡ L in + P | a k-1 -a k-1 | ≥ ε,
P k j=1 ζ Lj n - k j=1 E ∆ j ≥ ε ≤ k j=1 P ∆ jn -E ∆ j ≥ ε 2k + P(Λ n ≥ q α n 2k ε).
The first term is handled with the help of the relation (A.1), it yields to a O(n -2 ε -4 ).

For the second term, Markov's inequality and the condition A4.1-(i) applied with p = 1 give the bound O(q -α n ε -1 ). Finally, we obtain a bound of order O(n -2 ε -4 ) + O(ε -1 q -α n ) + O(nq

-min(1,αp) n
). The rate is obtained for ε n = (log n) c n -1 4 , c > 1 4 , q n = n -β with β > max( 5 4α , 2 min(1,αp) ) and Borel Cantelli's lemma. Proof of Proposition 5.2. First, we state the following property.

Property A.1. Let (u j , j = 1, . . . , p) and (v j , j = 1, . . . , p) be positive numbers such that max The proof is concluded with the classical approximations of X i by Z i and L iσi(j)n /q n by T iσi(j) . Here, all the quantities are a.s. bounded so we may make use of the Hoeffding's inequality to derive the claimed exponential bounds. Details are left to the reader.

|u 1 u 2 -v 1 v 2 | = |u 1 (u 2 -v 2 ) + v 2 (u 1 -v 1 )| ≤ d |u 1 -v 1 | + |u 2 -v 2 | .

ζ

  iℓn ≥ ζ iLj n . First, the second term does not depend on ε and it is a O(nq -αp n ) + O(nq -1 n ) by Theorem 5.1. Next from (A.4), we may derive analogously to the equation (A.2):

  v j | , p ≥ 2. Proof of Property A.1. If p = 2,

ζ

  (A.6) Now, set α p-1 = u 1 • • • u p-1 and β p-1 = v 1 • • • v p-1 , then from (A.6) and by induction|α p-1 u p -β p-1 v p | ≤ α p-1 |u p -v p | + v p |α p-1 -β p-1 | ≤ d p-1 |u p -v p | + d(d p-2 p-1 j=1 |u j -v j |) ≤ d p-1 p j=1 |u j -v j | .Hence the result.Next for proving Proposition 5.2, we begin as in the proof of Proposition 5.1-(2). Setting ζ ijn = X iσi (ℓp)n -E ∆ σ(ℓp) .

  in Theorem 4.2, an expected rate of convergence to estimate the jumps' intensities is O (log n) γ n -1 2 .

	4.3. Estimation of the maximal jump. Suppose that there exists a unique
	integer k max such that E (∆ 1kmax ) > max j=1,...,k	E (∆ 1j ). Then, an estimator of the
				j =kmax
	maximal intensity of jump is ∆ max = max j=1,...,kn	∆ j with k n → ∞ as n → ∞. From
	max j=1,...,kn	∆ j -E (∆ 1j ) ≥	max j=1,...,kn	| ∆ j | -max j=1,...,kn	|E (∆ 1j )| , we get that for all
	ε > 0:

  The desired probability is clearly bounded by

	n	k
	i=1	j=1	P max ℓ=1,...,qn ℓ ∈Lin

Acknowledgments. We want to thank the Reviewers and the Managing Guest Editor for improving the first version of this paper.

2) a

4 < ∞, j = 1, . . . , k and q n = n β with β > max( 5 4α , 2 min(1,αp) ). Note that if Assumption A4.2-(i) is fulfilled with p ≥ 2, the condition β > max( 5 4α , 2 min(1,αp) ) may be reduced to β > max( 5 4α , 5 2 ). Finally to simplify the study of our estimators, we add an assumption of boundedness and derive the following result for the coefficients a k-j , j = 2, . . . , k.

Proposition 5.2. Under the hypotheses of Proposition 5.1, we suppose in addition that X ≤ C and that functions a, b and M i are bounded. Then, for j = 2, . . . , k and

The special case of k = 2 and conclusion. Collecting the previous results, we obtain that a k-j a.s.

----→ n→∞ a k-j for each j = 1, . . . , k. Now the problem consists in solving the equation k j=0 a k-j,n x j = 0 (with a k,n ≡ 1) to recover the roots E (∆ σ(j) ), j = 1, . . . , k. For k = 2, the resolution is straightforward and gives the solutions

. We easily derive the strong consistency of these estimators with the help of the propositions 5.1 and 5.2. The cases k = 3 and k = 4 are again rather easy to handle but for k > 4, the use of numerically approximated solutions should be considered. Simulations should be carried out to see how estimation is involved in the accuracy of this approximation.

Appendix A. Auxiliary proofs

The proofs of Lemmas 3.2 and 3.3 being similar, we only give the derivation of the latter one.

Proof of Lemma 3.3.

1) Suppose that the conditions given in A3.2-(i) are fulfilled. Similarly to the proof of Theorem 1 p. 388-389 in [START_REF] Shiryaev | Jump-diffusion models: a practitioners guide[END_REF], we get that

for independent and centered random variables

| are independent with finite fourth moment thanks to the condition A3.2-(i). For the term P Λ n > ηq α n , we apply the Markov's inequality and get for n large enough that this term is a O(q -α n ).

1 Note that concerning the continuous framework, this case appears in [START_REF] Blanke | Exponential bounds for intensity of jumps[END_REF] with a slight misstatement since it is necessary to assume independence between the two jumps for K = 2.

2) If the conditions A3.2-(ii) are fulfilled, exponential moments do exist and we have a(X 1 ) < a ∞ , so we may use Bernstein inequality to get the claimed exponential bound. Concerning the term involving Λ n : we first bound it with

Next, since the b(Z i ) and M i are independent random variables with exponential moment, we obtain by Markov's inequality that these two terms are of order O e -c nq α n for some c > 0.

Proof of Theorem 3.2.

We start from the relation (3.10) with three terms to study. The first one, P( k = k), is controlled in Theorem 3.1. For the second term, we set η = pj 2 and ε 1 = εpj 4 , so for j = 1, . . . , k

The term P( ℓ j = ℓ j ) is also controlled with Theorem 3.1. Next, from a similar bound as in (3.7), the first probability is bounded by

Following the beginning of the proof of Lemma 3.3, the condition A3.2-(i) gives the bounds O n -2 ε -4

prevailing those obtained in (3.2) for P( ℓ j = ℓ j ) and P( k = k) as soon as ε 1 u -1 n → 0. On the other hand, under A3.2-(ii) and following the second part of Lemma 3. .


Francky Fouedjio Kameni
email: francky.fouedjiokameni@csiro.au

Serge Séguret

Breccia Pipe Prediction: a new approach using non-stationary covariances

In this paper, we are interested in the prediction of the breccia pipe elevation named "Braden" of the El Teniente mine in Chile. The problem is tackled by developing a new geostatistical approach based on non-stationary covariances. The proposed method offers an integrated treatment of all aspects of non-stationarity: mean, variance and spatial continuity. The estimation of non-stationary parameters is free distribution and carried out under the local stationarity assumption. The resulting estimated non-stationary parameters are naturally integrated into a kriging procedure or conditional simulations. The proposed approach has revealed an increased prediction accuracy when compared to the stationary one and demonstrated the ability to extract the underlying non-stationarity.

Introduction

A canonical problem in the geosciences is the prediction of a physical quantity over the whole region of interest from a finite set of irregular spaced data. This problem involves modeling and estimating the underlying spatial dependence structure of the observed data. Commonly, this is accomplished through statistical tools such as the variogram or covariogram computed on the whole domain of interest, under the stationarity assumption. However, in practice the stationarity assumption can be doubtful due to many factors, including specific landscape and topographic features of the region of interest or other localized effects. These local influences can be reflected computing local variograms, whose characteristics may vary across the domain of study. In such cases, carry out predictions based on a stationary approach could produce less accurate predictions, including an incorrect assessment of the estimation error [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF].

Several approaches have been proposed for modeling and estimating non-stationary dependence structure (see [START_REF] Guttorp | Covariance Structure of Spatial and Spatio-temporal Processes[END_REF], for a brief review ). One of the most interesting is the explicit non-stationary covariances class proposed by [START_REF] Paciorek | Spatial Modelling Using a New Class of Nonstationary Covariance Functions[END_REF]. However, the parameter estimation of these latter remains a crucial problem. In this work, we develop a procedure of estimating parameters that govern this class of closed-form non-stationary covariances under a single realization and local stationarity framework, through a step by step approach. First, we compute local variograms by a non-parametric kernel estimator. Then, it is used in a weighted local least squares procedure for estimating the parameters at a reduced set of representative points referred to as anchor points. Finally, a kernel smoothing method is used to interpolate the parameters at any location of interest. Then, the estimated non-stationary parameters are integrated naturally into a kriging procedure or conditional simulations.

As a motivating example in this work we consider the prediction of the breccia pipe elevation named "Braden" of the El Teniente mine in Chile. This latter is one of the largest known porphyry-copper ore bodies. The pipe is poorly mineralized and surrounded by different kinds of mineralized geological units. Knowing the exact location of the pipe surface is important, as it constitutes the internal limit of the deposit. Previous approaches have been applied on this dataset by [START_REF] Séguret | Geometric Modeling of a Breccia Pipe -Comparing Five Approaches[END_REF].

The paper is structured as follows: the model formulation is described in Section 2. In Section 3, the statistical inference is detailed. Spatial predictions and conditional simulations are presented in Section 4. In Section 5 the proposed approach is applied on the breccia pipe datasets. Section 6 is a concluding remarks.

Model Formulation

Let Y = {Y (x) : x ∈ G ⊆ R p , p ≥ 1} be a random field defined on a fixed continuous domain of interest G of the Euclidean space R p and reflecting the underlying studied phenomenon. We consider that Y is governed by the following model:

Y (x) = m(x) + σ(x)Z(x), ∀x ∈ G, (1)
where: m : R p → R is an unknown fixed function; σ : R p → R + is an unknown positive fixed function; Z is a zero-expectation, unit variance random field with correlation function defined by:

R N S (x, y) = φ xy R S Q xy (x -y) , ∀(x, y) ∈ G × G, (2)
with:

φ xy = |Σ x | 1 4 |Σ y | 1 4 Σ x + Σ y 2 -1 2 , Q xy (h) = h T Σ x + Σ y 2 -1 h, ∀h ∈ R p ; Σ : R p → P D p (R), x → Σ
x is a mapping from R p to P D p (R) the set of real-valued positive definite p-dimensional square matrices; R S (.) is a continuous isotropic stationary correlation function, positive definite on R p , for all p ∈ N .

The expression (2) represents the closed form non-stationary covariances class proposed by [START_REF] Paciorek | Spatial Modelling Using a New Class of Nonstationary Covariance Functions[END_REF]. The construction of this class is based on a convolution of an orthogonal random measure with a spatially varying random weighting function (see [START_REF] Fouedjio | A Generalized Convolution Model and Estimation for Non-stationary Random Functions[END_REF] for more details). We can notice that this class gives non-stationary versions of some well known stationary correlation functions (Gaussian, exponential, Matérn and Cauchy) for a specific choice of R S (.). The intuition behind this class is that to each location x is assigned a local Gaussian kernel matrix Σ x and the correlation between two locations x and y is calculated by averaging between the two local kernels at x and y. In this way, the local characteristics at both locations influence the correlation of the corresponding target values. Thus, it is possible to account for non-stationarity. It is done by specifying the mapping Σ(.) which models the anisotropy of the correlation function. The resulting kernel matrix Σ x at each point x is interpreted as a locally varying geometric anisotropy matrix. It controls the anisotropic behavior of the random field in a small neighborhood around x.

From model defined in (1), the two first moments of the random field Y is given by:

E(Y (x)) = m(x), (3)
Cov(Y (x), Y (y)) = σ(x)σ(y)R N S (x, y) ≡ C N S (x, y). (4)
Then, the non-stationarity of the random field Y is characterized by the non-stationary parameters m(.), σ(.) and Σ(.) defined at any location of the region of interest.

Statistical Inference

Let Y = (Y (s 1 ), . . . , Y (s n )) T be a (n × 1) vector of observations from a unique realization of the random field Y , associated to known locations {s 1 , . . . , s n } ⊂ G ⊆ R p . The objective is to use the data Y to estimate the mean function m(.), the standard deviation function σ(.) and the correlation function determined by Σ(.). The estimation of these parameters relies on the slightly local stationarity assumption which allows certain simplifications.

Local Stationarity

The local stationarity assumption [START_REF] Matheron | The Theory of Regionalized Variables and its Applications[END_REF] implies that at any location x 0 ∈ G there exists a neighborhood V x 0 = {x ∈ G, xx 0 ≤ b} where the random field Y can be approximated by a stationary randon field. Thus, ∀(x, y) ∈

V x 0 × V x 0 , m(x) ≈ m(y) ≈ m(x 0 ) and C N S (x, y) ≈ C S (x -y; x 0 ) = C S (h; x 0 ), h ≤ b;
where C S (.) is a stationary covariance and the limit b represents the radius of the local stationarity neighborhood V x 0 . In this way, the parameters are assumed to be very smooth functions which vary slowly over the domain. The expectation of the random field Y being approximately equal to a constant inside the local stationarity neighborhood, the resulting local covariance structure at any location x 0 is written as follows:

C S (h; x 0 ) = σ 2 (x 0 )R S h T Σ -1 x 0 h , h ≤ b. (5)
Locally, the non-stationary covariance C N S (., .) (4) is thus reduced to an anisotropic stationary one C S (.) (5). The anisotropy function Σ(.) is parametrized through the spectral decomposition and then the positive definiteness is guaranteed. Precisely, at any location

x 0 ∈ G, Σ x 0 = Ψ x 0 Λ x 0 Ψ T
x 0 , where Λ x 0 is the diagonal matrix of eigenvalues and Ψ x 0 is the eigenvector matrix. We assume working in 2D (p = 2) from now and we have:

Λ x 0 = λ 2 1 (x 0 ) 0 0 λ 2 2 (x 0 ) , Ψ x 0 = cos ψ(x 0 ) sin ψ(x 0 ) -sin ψ(x 0 ) cos ψ(x 0 ) , λ 1 (x 0 ), λ 2 (x 0 ) > 0 and ψ(x 0 ) ∈ [0, π).
At each point, the square roots of the eigenvalues control the local ranges and the eigenvector matrix specify the local orientations. Thus, the anisotropy function Σ(.) is characterized by the functions λ 1 (.), λ 2 (.) and ψ(.).

Local Variogram Kernel Estimator

Under the local stationarity assumption, we define a non-parametric kernel moment estimator of the stationary local variogram at a fixed location x 0 ∈ G and lag h ∈ R p , γ(h; x 0 ) = σ 2 (x 0 ) -C S (h; x 0 ), h ≤ b as follows:

γ (h; x 0 ) = V (h) K (x 0 , s i )K (x 0 , s j )[Y (s i ) -Y (s j )] 2 2 V (h) K (x 0 , s i )K (x 0 , s j ) , h ≤ b, (6)
where the average ( 6) is taken over V (h) = {(s i , s j ) : s i -s j = h}, the set of all pairs of locations separated by vector h; K (x 0 , s i ) = K (x 0 , s i )/ n l=1 K (x 0 , s l ) are standardized weights; K (., .) is a non-negative, symmetric kernel on R p × R p with bandwidth parameter > 0.

This moment estimator of the local variogram at any location x 0 ∈ G is a kernel weighted local average of squared differences of the regionalized variable. The kernel function is used to smoothly down-weight the squared differences (for each lag interval) according to the distance of these paired values from a target location. We assign to each data pair a weight proportional to the product of the individual weights. Observation pairs near to the target location x 0 have more influence on the local variogram estimator than those which are distant.

To calculate the non-parametric kernel estimator (6), we choose an isotropic stationary Gaussian kernel:

K (x, y) ∝ exp(-1 2 2 x -y 2 ), ∀(x, y) ∈ G × G.
The latter has a noncompact support and therefore considers all observations. Thus, the local variogram estimator is not limited only to the local information, distant points are also considered. This avoids artefacts caused by the only use of observations close to the target location. It also reduces instability of the obtained local variogram at regions with low sampling density. Furthermore, it provides a smooth parameter estimate and then is compatible with the quasi-stationarity assumption. The size of the quasi-stationarity neighborhood b, it is set with respect to the bandwidth . We take b = √ 3 such that the standard deviation of the isotropic stationary Gaussian kernel matches the isotropic stationary uniform kernel (with compact support).

Parameter Estimation

The estimation of the parameters vector θ(x 0 ) = (σ(x 0 ), λ 1 (x 0 ), λ 2 (x 0 ), ψ(x 0 )) which characterizes the stationary local variogram γ(.; x 0 ) ≡ γ(.; θ(x 0 )) at a fixed location x 0 are found via the following minimization problem:

θ(x 0 ) = arg min θ(x 0 )∈Θ w (x 0 ) (γ(θ(x 0 )) -γ (x 0 )) , (7)
where is the product term by term ; γ T (θ

(x 0 )) = [γ(h j )] j=1...J ; γ T (x 0 ) = [ γ (h j ; x 0 )] j=1...J ; w T (x 0 ) = [w (h j ; x 0 )] j=1...J , w (h; x 0 ) = ( V (h) K (x 0 , s i )K (x 0 , s j ))/ h 1/2
; {h j ∈ R p , j = 1, . . . , J} are given lag vectors; θ(x 0 ) ∈ Θ is the vector of unknown parameters and Θ is an open parameter space.

Using the estimate of the vector of structural parameters θ(x 0 ) obtained in ( 7), the mean parameter m(x 0 ) is estimated explicitly by a local stationary kriging of the mean [START_REF] Matheron | The Theory of Regionalized Variables and its Applications[END_REF].

For the prediction purpose, one needs to compute the non-stationary parameters m(.), σ(.) and Σ(.) at prediction and observation locations. In practice, it is unnecessary to solve the minization problem (7) at each target location. Indeed, so is computationally intensive and redundant for close locations, since these estimates are highly correlated. To reduce the computational burden, the proposed idea consists in obtaining the parameter estimates only at some reduced set of m n representative points referred to as anchor points defined over the domain of interest. Then, using the estimates obtained at anchor points, a kernel smoothing method is used to make available estimates at any location of interest. We works with the Nadaraya-Watson kernel smoother which is appropriate and relatively simple. However, other smoothers can be used as well (local polynomials, splines, etc.). The choice of the smoothing bandwidth associated to Nadaraya-Watson kernel smoother is done through the generalized cross-validation criteria [START_REF] Wand | Kernel Smoothing[END_REF].

The estimation of the non-stationary parameters depends on the bandwidth parameter used in the computation of the local variogram non-parametric kernel estimator defined in (6). Indeed, the size of the local stationarity neighborhood is expressed in terms of this bandwidth parameter. The data-driven method used to select the bandwidth consists of leaving out one data location and using a form of cross-validation. Because the estimation of the spatial dependence structure is rarely a goal per se but an intermediate step before kriging, we want to choose the bandwidth that gives the best cross-validation mean square error.

Prediction

The main purposes of modelling and estimating the spatial dependence structure is to spatially interpolate data and perform conditional simulations. The expected benefit using the closedform non-stationary covariances class ( 4) is to obtain spatial predictions and variance estimation errors more realistic than those based on a inadequate stationary covariances.

Kriging

Let C N S (., .) the non-stationary covariance of the random field Y and m(.) its mean. Given the vector of observations Y = (Y (s 1 ), . . . , Y (s n )) T at n fixed locations s 1 , . . . , s n ∈ G, the point predictor for the unknown value of Y at unsampled location s 0 ∈ G is given by the optimal linear predictor:

Y (s 0 ) = m(s 0 ) + n i=1 η i (s 0 )(Y (s i ) -m(s i )). ( 8
)
The kriging weight vector η = [η i (s 0 )] and the corresponding kriging variance Q(s 0 ) are given by:

η = C -1 C 0 et Q(s 0 ) = σ 2 (s 0 ) -C T 0 C -1 C 0 . ( 9
)
where

C 0 = [C N S (s i , s 0 )]; C = [C N S (s i , s j )].

Conditional Simulations

Here we assume that the random field Y is Gaussian with mean m(.) and non-stationary covariance structure C N S (., .). We want to simulate at a large number of locations a Gaussian random field with same mean and covariance, and ensure that the realization honors the observed values Y (s 1 ), . . . , Y (s n ). This can be achieved from an unconditional simulation of the random field Y as follows [START_REF] Lantuejoul | Geostatistical Simulation: Models and Algorithms[END_REF]:

1. realize a unconditional simulation {X(s), s ∈ G} of the random field Y ;

2. carried out a simple kriging of {X(s) -Y (s), s ∈ G} from its values taken at the data points {s i , i = 1, . . . , n}, using m(.) and C N S (., .) ; 3. add the unconditional simulation and the result of kriging.

We have

Y (x) = m(x) + σ(x)Z(x), ∀x ∈ G,
where Z is a Gaussian random field with zero expectation, unit variance and non-stationary correlation function R N S (., .). Thus, to simulate the Gaussian random field Y (step 1 of the previous algorithm), we need to known how we can simulate Z. Simulation of the Gaussian random field Z can be carried out using a propagative version of the Gibbs sampler proposed by [START_REF] Lantuejoul | Simulation of a Gaussian Random Vector: A Propagative Version of the Gibbs Sampler[END_REF]. This algorithm allows to simulate a Gaussian vector at a large number of locations (comparatively to the existing classical algorithms such as Cholesky method or Gibbs sampler) without relying on a Markov assumption (it does not need to have a sparse precision matrix). The algorithm proposed in [START_REF] Lantuejoul | Simulation of a Gaussian Random Vector: A Propagative Version of the Gibbs Sampler[END_REF] requires neither the inversion nor the factorization of a covariance matrix. Note that simulation methods such as spectral method or turning bands method are not adapted to the non-stationary case [START_REF] Lantuejoul | Geostatistical Simulation: Models and Algorithms[END_REF]. The representation that underlies these methods relies on the stationarity assumption.

Application

The methodology presented in Section 4 has been applied to the elevation data of the breccia pipe called "braden" of the El Teniente mine in Chile. We have a training data (616 observations) which serves to calibrate the model and a validation data (200 observations) which serves only to assess the prediction performances. A comparison scheme of kriging under stationary and non-stationary models is carried out through a validation sample.

Raw estimates of non-stationarity parameters m(.), σ 2 (.) and Σ(.) at anchor points are shown respectively on Figures 1b, 1c and1d. They are based on the non-stationary exponential covariance function. Concerning the estimated anisotropy function Σ(.) at anchor points, it is represented by ellipses as shown in Figure 1d. Based on these estimates, non-stationarity in the data is quite visible. Especially, from Figure 1d where we can clearly see the spatially varying azimuth. Such directional effects are also quite apparent on data (Figure 1a). Note that the stationary approach has not detected a global geometric anisotropy. Figure 2 shows the maps of smoothed parameters over the whole domain of observations: mean, variance, anisotropy ratio and azimuth. A visualization of the covariance at certain points (with all other points) via the level contours for estimated stationary and non-stationary models is presented in Figure 3. We can see how the non-stationary spatial dependence structure changes the shape from one place to another as compared to the stationary one. The stationary model is a nested isotropic model (nugget effect, exponential and spherical) while the nonstationary model corresponds to the non-stationary exponential covariance function. Table 1 presents the summary statistics for the external validation (200 hold-out sample) results using the classical stationary approach and the non-stationary proposed one. Some well-known discrepancy measures are used [START_REF] Chilès | Geostatistics: modeling spatial uncertainty[END_REF]), namely the Mean Absolute Error (MAE), the Root Mean Square Error (RMSE), the Normalized Mean Square Error (NMSE), the Logarithmic Score (LogS) and the Continued Rank Probability Score (CRPS). For RMSE, LogS and CRPS, the smaller the better; for MAE, the nearer to zero the better; for NMSE the nearer to one the better. Table 1 shows that the proposed approach outperforms the stationary one with respect to all the measures. The cost of non-using the non-stationary approach in this case is substantial: in average the prediction at validation locations is about 23% better for the non-stationary approach than for the stationary one, in terms of RMSE. The kriged values and the kriging standard deviations for the estimated stationary and nonstationary models are shown in Figure 4. The overall look of the predicted values and prediction standard deviations associated with each model differ notably. In particular, the proposed method takes into account certain local characteristics (such as locally varying anisotropy) of the regionalization that the stationary approach is unable to retrieve. Figure 5 shows some conditional simulations in the Gaussian framework, based on the estimated non-stationary model.

Conclusion

In this paper we are proposed a statistical methodology based on a non-stationary covariances class to predict the elevation of the breccia pipe elevation named "Braden" of the El Teniente mine in Chile. The estimation method offers an integrated treatment of all aspects of non-stationarity (mean, variance, covariance) in the modeling process and relies on the mild hypothesis of quasi-stationarity. The proposed method has revealed an increased prediction accuracy when compared to the standard stationary method, and demonstrated the ability to extract the underlying non-stationarity from a single realization. It also provides an exploratory analysis tool for the non-stationarity. Beyond the spatial predictions, we also show how conditional simulations can be carried out in this non-stationary framework.

Figure 1 :

 1 Figure 1: (a) Training data; (b) Estimated mean function m(.) at anchor points; (c) Estimated variance function σ 2 (.) at anchor points; (d) Estimated anisotropy function Σ(.) at anchor points where the ellipses were scaled to ease vizualisation.

Figure 2 :Figure 3 :

 23 Figure 2: Smoothed parameters over the domain of observations: (a) mean, (b) variance, (c) anisotropy ratio, (d) azimuth.

Figure 4 :

 4 Figure 4: (a,b) Predictions and prediction standard deviations for the estimated stationary model. (c,d) Predictions and prediction standard deviations for the estimated non-stationary model.

Figure 5 :

 5 Figure 5: Conditional simulations based on the estimated non-stationary model.

Table 1 :

 1 External validation on a set of 200 observations.

		Stationary Non-stationary
	MAE	79.74	61.39
	RMSE	154.41	117.99
	NMSE	0.98	0.74
	LogS	2439	2315
	CRPS	123.59	121.11

Acknowledgements

The authors would like to thank the company Codelco, Chile for providing the data used in this paper.