Exponential inequalities for the supremum of some counting processes and their square martingales

We establish exponential inequalities for the supremum of martingales and square martingales obtained from counting processes, as well as for the oscillation modulus of these processes. Our inequalities, that play a decisive role in the control of errors in statistical procedures, apply to general non-explosive counting processes including Poisson, Hawkes and Cox models. Some applications for U -statistics are discussed.


Introduction
Counting processes naturally arise in a lot of applied fields and the understanding of their evolution is the object of a lot of modelling problems. In this context, exponential inequalities are of great interest, especially for the control of errors in statistics. Exponential inequalities for the distribution of random variables have been of interest for many years (see Hoeffding [1963] for one of the first result in this field), and they are still a very active research area for various types of processes, like sums of i.i.d. random variables, empirical processes, U -statistics, Poisson processes, martingales and self-normalised martingales, with discrete or continuous time. For example, for discrete time processes with i.i.d. random variables, exponential inequalities have been obtained for the empirical process or for U -statistics of order two in Hanson and Wright [1971], Giné and Zinn [1992], Arcones and Giné [1993], Talagrand [1996], Ledoux [1997], Klass and Nowicki [1997], Bretagnolle [1999], Massart [2000] or Giné et al. [2000] to cite a few. We may refer also to Massart [2007] or Bercu et al. [2015] for a wide review of exponential inequalities for discrete time martingales.
In this paper we focus on counting processes in continuous time. Our aim is to provide exponential inequalities with explicit constants for general counting processes, specifically for their associated local martingales, for their local square martingales and for their oscillation modulus. The first one is useful for statistical applications like density estimation as exploited for instance in Reynaud-Bouret [2003]. The second one involving local square martingales allows to control U-statistics (see Houdré and Reynaud-Bouret [2003]) which have a long history. For instance, the estimation of a quadratic functional of a density (Laurent [2005]), or in testing problems (see Fromont and Laurent [2006] for a goodness-of-fit test in density or Fromont et al. [2011] for an adaptive test of homogeneity of a Poisson process), the estimator, as well as the test statistics are naturally U -statistics of order two. As to the third contribution concerning the oscillation modulus, applications may concern multiple testing problems where some procedures are based on the oscillation modulus of counting processes, which entails the need to control the supremum and the whole oscillation modulus of these processes, and not only their marginals. In a non-asymptotic framework, it is necessary to obtain inequalities with explicit constants. The keystone for controlling the statistical error is then to use exponential bounds for the right model, but the results obtained on square martingales are generally not the simple consequence of those obtained for simple martingales. To achieve our goal, we first exhibit local martingale properties of the exponential of counting processes, then we state exponential inequalities for the supremum of those processes, leading to exponential bounds for the oscillation modulus.
A first exponential inequality for martingales of counting processes in continuous time can be found in Theorem 23.17 of Kallenberg [1997], that concerns semimartingales M under the restrictive assumption that [M ] ∞ ≤ 1 almost surely. The specific case of the Poisson process is studied in Reynaud-Bouret [2003]. More general counting processes are considered in Van De Geer [1995] or Reynaud-Bouret [2006]. In these contributions, the exponential bounds are derived from techniques adapted from the empirical process, with extensions of Bernstein's exponential inequality for general martingales. As a consequence of the results of Van De Geer [1995], exponential inequalities with explicit constants have been established in Reynaud-Bouret [2006] for the supremum of counting processes with absolutely continuous compensators, as well as for the supremum of a countable family of martingales associated with counting processes.
However the case of the square martingale is not addressed in these previous results. As to the existing results concerning exponential inequalities for U-statistics, many of them come from results on sequences of i.i.d. random variables. Indeed, in the specific case of the Poisson process, a sharp exponential inequality with explicit constants holds for U -statistics of order two and for double integrals of Poisson processes in Houdré and Reynaud-Bouret [2003]. The Poisson process is seen as a point process (T i ) i≥1 on the real line, allowing to use the inequalities obtained for U -statistics of i.i.d. random variables like Rosenthal's inequality and Talagrand's inequality, after conditioning by the total random number of point. Unfortunately this approach is no longer valid when we consider more general counting processes than the Poisson process.
This contribution unifies the above approaches since our exponential inequalities apply to general counting processes (including the Poisson process, non-explosive Cox and Hawkes processes under mild assumptions like bounded intensities for instance), and we consider both martingales and squaremartingales. Comparing to the existing literature, we do not make any assumption about the independence of the underlying point process and we use quite different proofs involving stochastic calculus instead of adapting previous techniques in discrete time. Moreover, we get sharper inequalities when applied to the setting of existing works. For instance concerning the exponential inequality for the martingale, we get a non-asympotic inequality with explicit constants and we obtain a tail of order x log(x) instead of x in Van De Geer [1995]. As an application of our results, we obtain a control of the supremum of some U -statistics and double integrals based on other counting processes than the Poisson process. We also provide an inequality for the oscillation modulus, which allows a fine control of quadratic statistics based on counting processes.
The remainder of this article is organized as follows: in the next section, we introduce some general notations, while Section 3 is devoted to the exponential martingales of counting processes. The exponential inequalities of our martingales and their associated square martingales are presented in Section 4, while Section 5 details applications to U-statistics and oscillation modulus. Finally, we have gathered all the proofs in Section 6.

Notations
Let (Ω, F, P) be a filtered probability space where F = (F t ) t≥0 is a complete right-continuous filtration, N = (N t ) t≥0 be a non-explosive F-adapted counting process whose jump times are totally inaccessible, and Λ = (Λ t ) t≥0 be its F-compensator. We consider H = (H s ) s≥0 , a bounded predictable process, bounded by the non-random real number H ∞, [c,d] [c,d] almost surely. If c = 0 and T ≥ 0, H ∞,[0,T ] will be written H ∞,T for short. The non-random real number H 2, [c,d] will stand for a bound of the Recall that for a semimartingale X, we define [X] t by where < X c > is the quadratic variation of the continuous martingale part of X and ∆X s = X s − X s − is the jump of X at s. We will use the fact that if X is a local martingale with jumps bounded by 1 and H is a bounded predictable process, then ( [Protter, 2005, Corollary 3 p.73]). Recall also that for a C 2 function f and a càdlàg semimartingale (Y t ) t≥0 , the Itô formula ( [Bass, 2011, Theorem 17.10 For f = exp and a semimartingale Y satisfying < Y c >≡ 0 and Y 0 = 0, this leads to Finally we define for every n ≥ 1 [Bass, 2010, Theorem 2.4]) satisfying lim n→+∞ S n (Y ) = +∞ almost surely.

Martingale properties
Let T > 0. We consider in this section the three processes M = (M t ) t≤T ,M = (M t ) t≤T and and By definition N − Λ is a local martingale. Since the jumps of N are totally inaccessible, we know that Λ is continuous and N − Λ has jumps bounded by 1. The local martingale N − Λ is then a locally square integrable local martingale of finite variations. As a consequence, M is a local martingale of finite variations, andM, as well as ≈ M, is a semimartingale of finite variations. Our main goal is to establish in the next section some exponential inequalities for these three semimartingales. We will use Chernoff's bounds in order to do that, so we are first interested by some exponentials associated with the three processes M,M and ≈ M. We start first with the process M in the following lemma, proving that the exponential of M is a local martingale. We follow the proof of Theorem VI.2 in Brémaud [1981] where the case of an absolutely continuous compensator Λ is treated. We may also refer to Sokol and Hansen [2012] to find in that case some conditions on the counting process and its intensity to obtain an exponential which is a martingale.
Lemma 1. Let Z be the process defined for a fixed real number λ and all t ≤ T by Then for every n ≥ 1, the process (exp(Z t∧Sn(Z) )) t≤T is a martingale.
Let us define now for a > 0 Since the jumps of N are totally inaccessible, T a is a stopping time ( [Bass, 2011, Proposition 16.3]). As a consequence of Lemma 1, replacing H by the process 2M s − H s 1 s≤Ta which is also a bounded predictable process and using (2), we obtain the next lemma which sets out a stopped martingale associated with the exponential ofM.
Lemma 2. LetZ be the process defined for a fixed real number λ and all t ≥ 0 bỹ For every positive a and every n ≥ 1, the process (exp(Z t∧Ta∧Sn(Z) )) t≥0 is a martingale.
Finally we present the analogue of Lemma 2 for the process ≈ M, which is a consequence of (3) and Lemma 1.
Lemma 3. Let ≈ Z be the process defined for a fixed real number λ and all t ≥ 0 by For every positive a and every n ≥ 1, the process (exp( )) t≥0 is a martingale.

Exponential inequalities
We have gathered in this section our main results, that is the exponential inequalities for the three processes M ,M and ≈ M. The rates that appear in these inequalities are governed by the rate function I defined for x ≥ 0 by We start with a technical lemma that provides useful properties for the proofs of the main theorems.
Lemma 4. Let I t (H, λ) be defined for t ≥ 0 by t 0 (e λHs − 1 − λH s )dΛ s . For t ≤ T and every real λ, we get the almost sure inequality where g(x) = e x − 1 − x. Moreover the function g satisfies for every positive A, B and x We present now in Theorem 1 an exponential inequality for the local martingale M , with its two-sided version.
Theorem 1. For every positive x and T , we have the following inequalities: and P( sup Such exponential inequalities have already been obtained for martingales with bounded jumps in Kallenberg [1997], Van De Geer [1995] and Reynaud-Bouret [2006]. In Kallenberg [1997], the bound is of the form exp(− Ax 2 1+Bx ) for some constants A and B, and is available for a semimartingale M such that [M ] ∞ ≤ 1 almost surely, which is not our case here. In Van De Geer [1995], the bound is of the form A exp(−Bx) for some constants A and B and x large enough. Finally in Reynaud-Bouret [2006], the inequality is of the form P(sup t∈[0,T ] sup a M a t ≥ A √ x + Bx) ≤ exp(−x) for a countable family of martingales (M a t ) t≥0 . Comparing to all these results, in the case of the large deviations, that is when x tends to infinity, (6) and (7) provide a sharper bound with a more accurate tail, namely in x log x instead of x. When x tends to zero, these bounds are similar (up to constants), taking the form A exp(−Bx 2 ).
The next Theorem deals with the square martingaleM. The same inequality is obtained for −M , leading to a two-sided inequality.
Theorem 2. For every positive x and T , we have the following inequalities: and thereby we have the following two-sided exponential inequality: If we compare (7) and (10), we can notice that the upper bound in (10) involves √ x instead of x in the inequality (7), leading to different bounds when x tends to zero, contrary to the case of the large deviations. Finally the next Theorem 3 is the analogue of Theorem 2 for the martingale ≈ M.
Theorem 3. For every positive x and T , we have the following inequalities: and thereby we have the following two-sided exponential inequality: Comparing now (7) and (13), we observe that M and ≈ M are behaving in the same way for x tending to zero. When x tends to infinity, (13) provides a similar bound (up to a constant) to (10), which is quite surprising in view of the relationship ≈ M =M + H 2 d(N − Λ). Moreover this relationship, (7) with H 2 instead of H, all along with (10) and x 2 , will also lead to an exponential inequality but less sharp than (13) because H 4 2,T ≤ H 2 2 2,T .

U-statistics of order two
The main hypothesis of the previous theorems is to suppose that the countable process is non-explosive with totally inaccessible jumping times. This allows us to consider for instance Poisson, Cox or Hawkes processes with a bounded intensity. If N is a Poisson process, some sharp exponential inequalities have already been obtained in Houdré and Reynaud-Bouret [2003] for double stochastic integrals of the form Z t = t 0 where h is a (non-random) bounded Borel function. The Poisson process N is viewed as a point process (T i ) i≥1 , so that Z t is a U -statistic for the Poisson process: Z t = 0≤T i <T j ≤t g(T i , T j ) for some function g. We may then use the inequalities obtained for U -statistics after conditioning by the total random number of points, leading to a similar inequality as the one in Giné et al. [2000]. However, Z t takes the form of a U -statistics for any counting process N , and not only for the Poisson process.
In the particular case where h is a stochastic kernel of the form h(x, y) = H(x)H(y),M may be writtenM t = 2Z t , i.e. it is a double stochastic integrals or a U -statistics of order two. Although we are not limited to the Poisson case, by the Meyer theorem (see [Protter, 2005, page 104]), the jumps of a Poisson process are totally inaccessible so that we may apply Theorem 2. Comparing to Giné et al. [2000] or Houdré and Reynaud-Bouret [2003], where the supremum of (Z t ) t≥0 is not considered and h is not random, the inequality (8) provides sharper bounds for the large deviations with an additional log x in our inequality. Indeed in Giné et al. [2000] or Houdré and Reynaud-Bouret [2003], the bound is of the form L exp(− 1 L min( x 1/2 A 1/2 , x 2/3 B 2/3 , x C , x 2 D 2 )) for some explicit constants A, B, C, D and L. Such exponential inequalities for U -statistics are very useful for statistical applications. For instance the estimation of the L 2 norm f 2 (x)dx of the density of i.i.d. random variables via selection model is considered in Laurent [2005] and Fromont and Laurent [2006]. The estimator of a quadratic distance is naturally a U -statistics of order two and the exponential inequality of Houdré and Reynaud-Bouret [2003] is a main tool for the study of the property of the estimator. In the Poisson model too, as in Fromont et al. [2011] where the homogeneity is tested, the method is based on an approximation of the L 2 -norm of the intensity of the underlying Poisson process. Since our theorems in Section 4 apply to more varied counting processes, these quadratic form estimation procedures can be generalized to more general contexts than the Poisson framework.

Oscillation modulus control
The main theorems of Section 4 provide also an upper bound for the oscillation modulus of the three processes M,M and ≈ M. We consider c, d and x three non-negative real numbers, and the counting process N c (t) = N t+c − N c whose compensator is Λ c (t) = Λ t+c − Λ c . The following theorem gives upper bounds for the oscillation modulus of the processes M andM. As far as we know, this is the first time such an exponential inequality is stated for counting processes. An analogous inequality can be obtained for ≈ M by following the same way of proof.
Theorem 4. For every non-negative x, c and d, we have the following inequality for the oscillation modulus of M : x 2 )).
For the processM, we get the following exponential upper bound leading to the exponential inequality for the oscillation modulus ofM : x 8 )).
In view of Theorems 1 and 2, the previous inequalities show that considering the oscillation modulus instead of the processes M andM themselves does not affect the rates (in x) of the exponential bounds, but only changes the constants. We obtain in Theorem 4 explicit constants with respect to the integrand H as well as the interval [c, d], which may be useful for applications.

Proofs
Proof of Lemma 1 The process Z is defined as λM t − t 0 (e λHs − 1 − λH s )dΛ s where λ is a fixed real number. Z is a càdlàg semimartingale of bounded variations because H is bounded and Λ, as well as M, is of bounded variations. The continuity of Λ entails the equality ∆Z s = λH s ∆N s . We get then from (1) that For n ≥ 1, the stopping time S n (Z) is defined by S n (Z) = inf{t > 0, e Z t− ≥ n}. Then for every s ≤ S n (Z) ∧ T, e Z s− ≤ n. Moreover, for every t ≤ T, To conclude, the result follows from [N − Λ] = N and the inequality Proof of Lemma 4 Let s ≤ t ≤ T and λ ∈ R. We use the following inequality: that is Integrating with respect to dΛ s we obtain For the proof of (5), consider the function h defined for λ > 0 by Proof of Theorem 1 Recall that I t (H, λ) is defined by t 0 (e λHs −1−λH s )dΛ s . We define the process Z as in Lemma 1 by Z t = λM t − I t (H, λ) and the stopping time S n (Z) for n ≥ 1 by S n (Z) = inf{t > 0, e Z t− ≥ n}. Since (S n (Z)) n≥1 is a non-decreasing sequence of stopping times with lim n→+∞ S n (Z) = +∞ almost surely, the sequence (sup 0≤t≤T ∧Sn(Z) M t ) n≥1 is constant for n large enough. We then get by monotony Using Lemma 4 (4), we obtain for all λ > 0, x > 0 and n ≥ 1, Doob's maximal inequality and Lemma 1 then lead to for every λ > 0 with g(x) = e x − 1 − x, so taking the limit in n and the infimum in λ, we get by (5) that is (6). Applying this inequality with −H instead of H, we obtain also Then (7) follows from the inequality Proof of Theorem 2 Let us begin with the proof of (8). We defineZ as in Lemma 2 byZ t = λM t − I t (2HM, λ), thereby (S n (Z)) n≥1 is a sequence of non-decreasing stopping times such that lim n→+∞ S n (Z) = +∞ almost surely. We proceed then as in the proof of Theorem 1. For all positive λ, a and x Using the inequality (17), we get for t ≤ T and λ > 0 .