determinantal point processes (DPPs for short), that are an important example of negatively associated point processes. DPPs are a type of repulsive point processes that were first introduced by Macchi [ 32 ] in 1975 to model systems of fermions in the context of quantum mechanics. They have been extensively studied in Probability theory with applications ranging from random matrix theory to non-intersecting random walks, random spanning trees **and** more (see [ 26 ]). From a statistical perspective, DPPs have applications in machine learning [ 29 ], telecommunication [ 15 , 33 , 22 ], biology, forestry [ 30 ] **and** computational statistics [ 2 ]. As a first result, we relate the association property of a point process to its α-mixing properties. First introduced in [ 36 ], α-mixing is a measure of dependence between random variables, which is actually more popular than PA or NA. It has been used extensively to prove **central** **limit** theorems for dependent random variables [ 7 , 16 , 23 , 27 , 36 ]. More details about mixing can be found in [ 8 , 16 ]. We derive in Section 2 an important covariance inequality for associated point processes (**Theorem** 2.5 ), that turns out to be very similar to inequalities established in [ 18 ] for weakly dependent continuous random processes. We show that this inequality implies α-mixing **and** precisely allows to control the α-mixing coefficients by the first two intensity functions of the point process. This result for point processes is in contrast with the case of random fields where it is known that association does not imply α-mixing in general (see Examples 5.10-5.11 in [ 10 ]). However, this implication holds true for integer-valued random fields (see [ 17 ] or [ 10 ]). As explained in [ 17 ], this is because the σ-algebras generated by countable sets are much poorer than σ-algebras generated by continuous sets. In fact, by this aspect **and** some others (for instance our proofs boil down to the control of the number of points in bounded sets), point processes are very similar to discrete processes.

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Organization of the paper. This paper is organized as follows. In Section 2 we introduce the object of our study **and** recall some useful previous results. More precisely, we introduce our geometric framework in Section 2.1. In Section 2.2 we introduce various notations that will allow us to make sense of the combinatorics involved in our problem. The random measures we study are defined in Section 2.3. Finally, we state the Kac–Rice formulas for higher moments in Section 2.4 **and** we recall several results from [2] concerning the density functions appearing in these formulas. In Section 3, we prove our main result, that is the moments estimates of **Theorem** 1.12. Section 4 is concerned with the proofs of the corollaries of **Theorem** 1.12. We prove the Law of Large Numbers (**Theorem** 1.7) in Section 4.1, the **Central** **Limit** **Theorem** (**Theorem** 1.9) in Section 4.2 **and** the remaining corollaries (Corollaries 1.14 **and** 1.15) in Section 4.3.

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Obtaining convergence rates for the **Central** **Limit** **Theorem** is the most classical application of Stein’s method. However, when it comes to Wasserstein distances, standard results obtained using Stein’s method are usually restricted to a smoothed Wasserstein distance of order 1 (see e.g. Section 12 [ 11 ]). While the Stein kernel approach of [ 18 ] could be used to derive convergence rates for Wasserstein distances of any order, the resulting bounds would involve proper- ties of the Stein kernel which cannot be easily computed in practice. Using our result, we are able to derive convergence rates in the **Central** **Limit** **Theorem** for Wasserstein distances of order p ≥ 2 which only involves moments of the random variables considered. More precisely, if we consider i.i.d random vari- ables X 1 , ..., X n in R d with E[X 1 ] = 0 **and** E[X 1 X 1 T ] = I d admitting a finite

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In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg **central** **limit** **theorem** for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan **and** Louhichi (1999), a new **central** **limit** **theorem** is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these **central** **limit** theorems are provided. All the usual causal or non causal time series: Gaussian, associated, linear, ARCH(∞), bilinear, Volterra processes,. . ., enter this frame.

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CNRS & TELECOM ParisTech ‡§ , Univ. Pierre et Marie Curie ¶ , Univ. Paris Est k
Adaptive **and** interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distribu- tions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive **and** interacting MCMC algo- rithms. We establish a **Central** **Limit** **Theorem** for additive function- als of unbounded functions under a set of verifiable conditions, **and** identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algo- rithm (a simplified version of the equi-energy sampler) is presented to support our claims.

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The selfsimilar processes have been widely studied due to their applications as models for various phenomena, like hydrology, network traffic analysis **and** mathematical finance.
An interesting class of selfsimilar processes is given as limits of the so called Non- **Central** **Limit** **Theorem** studied in [21] **and** [9]. We briefly recall the context. Let g be a function of Hermite rank k (see Section 5 for the definition) **and** let (ξ n ) n∈Z be a stationary

We establish the first central limit theorem in dimension d in this class of problems for the length of the tree and matching produced by a natural heuristic (the exo[r]

THE MULTIVARIATE **CENTRAL** **LIMIT** **THEOREM**
T. O. GALLOUET, G. MIJOULE, **AND** Y. SWAN
Abstract. Consider the multivariate Stein equation ∆f − x · ∇f = h(x) − Eh(Z), where Z is a stan- dard d-dimensional Gaussian random vector, **and** let f h be the solution given by Barbour’s generator approach. We prove that, when h is α-Hölder (0 < α 6 1), all derivatives of order 2 of f h are α-Hölder up to a log factor; in particular they are β-Hölder for all β ∈ (0, α), hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For α = 1, the regularity we obtain is optimal, as shown by an example given by Raič [18]. As an application, we prove a near-optimal Berry-Esseen bound of the order log n/ √ n in the classical multivariate CLT in 1-Wasserstein distance, as long as the underlying random variables have ﬁnite moment of order 3. When only a ﬁnite moment of order 2 + δ is assumed (0 < δ < 1), we obtain the optimal rate in O(n − δ 2 ). All constants are explicit **and** their dependence on the dimension d is studied when d is large.

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All simulations were done for different sample sizes **and** different bootstrap samples, n **and** m n . The results are
presented in Tables 2, 3, 4 **and** 5, respectively.
We observe that the power of the test is very high in most cases. For the Exponential distribution, the power is close to 1. Indeed this distribution is very different from the Gaussian distribution since it is asymmetric, making it easy to discard the null assumption. The three other distributions do share with the Gaussian the property of symmetry, **and** yet the power of the test is also close to 1; it also increases with the sample size. Finally, for Student’s t distribution, the higher the number of degrees of freedom, the more similar it becomes to a Gaussian distribution. This explains

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t −→ t→∞ N (0, σ
2 ) ,
where N (0, σ 2 ) denotes usual Gaussian law. Such phenomena have also been
observed in the discrete setting of OQWs [1]. A key point to show this result is the use of the quantum trajectories associated to the CTOQWs. In general, quantum trajectories describe evolutions of quantum system undergoing indirect measurements (see [3] for an introduction). In the context of CTOQWs, quantum trajectories describe the evolution of the states undergoing indirect measurements of the position of the walker. In particular these quantum trajectories appear as solution of jump-type stochastic differential equations called stochastic master equations (see [24] for link between discrete **and** continuous time models in the context of OQW, one can also consult [5] for such an approach in the context of Open Quantum Brownian Motion). In the physic literature, note that such models appear also naturally in order to describe non-Markovian evolutions. They are called non-Markov generalization of Lindblad equations (see [6, 25, 4]).

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where (ε i,j ) are i.i.d. random variables with unknown distribution µ. The functions g j belong to a class G of deformation functions, which models how the distributions µj ’s can be warped one to another by functions in the chosen class. This model is the natural extension of the functional deformation models studied in the statistical literature for which estimation pro- cedures are provided in Gamboa, Loubes **and** Maza (2007) while testing issues are tackled in Collier **and** Dalalyan (2015). In the setup of warped distributions a main goal is the estimation of the warping functions, possibly as a first step towards registration or alignment of the (esti- mated) distributions. Of course, without some constraints on the class G the deformation model is meaningless (we can, for instance, obtain any distribution on R d as a warped version of a fixed probability having a density if we take the optimal transportation map as the warping function; see Villani (2009)) **and** one has to consider smaller classes of deformation functions to perform a reasonable registration. In the case of parametric classes estimation of the warping functions is studied in Agull´o-Antol´ın et al. (2015). However, estimation/registration procedures may lead to inconsistent conclusions if the chosen deformation class G is too small. It is, therefore, im- portant to be able to assess fit to the deformation model given by a particular choice of G **and** this is the main goal of this paper. We note that within this framework, statistical inference on deformation models for distributions has been studied first in Freitag **and** Munk (2005). Here we provide a different approach which allows to deal with more general deformation classes.

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In general, there always exists an optimal alignment (r 0 , r 1 , r 2 , ..., r d ) sat- isfying both ( 2.10 ) **and** ( 2.11 ) with, say, v = n α as above. (Consider any one of the longest common subsequences **and** choose the r i ’s so that these two con- ditions are satisfied.) Therefore, we slightly changed the framework of [ 13 ] as the argument below requires the existence of an optimal alignment with ( 2.11 ) for any value of X **and** Y . However, the proof of **Theorem** 2.3 proceeds as the proof of the corresponding result in [ 13 ], **and** is therefore omitted. (The only difference is that counting the cases of equality, an upper estimate on the number of integer-vectors (0 = r 0 , r 1 , . . . , r d−1 , r d = n) satisfying ( 2.10 ) is now given by

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Section 4 is devoted to applications. In particular, we give sufficient conditions for some functions of Harris recurrent Markov chains **and** for functions of linear processes to satisfy the bound (1.6) in the case (r, p) 6= (1, 3) **and** the rate O(n −1/2 log n) when r = 1 **and** p = 3. Since
projective criteria are verified under weak dependence assumptions, we give an application to functions of φ-dependent sequences in the sense of Dedecker **and** Prieur (2007). These conditions apply to unbounded functions of uniformly expanding maps.

STOCHASTIC RANDOM WALKS IN A RANDOM ENVIRONMENT.
TOMASZ KOMOROWSKI **AND** STEFANO OLLA
Abstract. We consider a class of two-fold stochastic random walks in a random environ- ment. The transition probability is given by an ergodic random field on Z d with two-fold stochastic realizations. The **central** **limit** **theorem** for this class of random walks has been claimed by Kozlov under certain strong mixing conditions (cf. [4], **Theorem** 3, p. 121). However the statement **and** the argument used in [4] are not correct, **and** we provide a coun- terexample in dimension two (cf. example 2.3 below). We give a sufficient condition for the walk to satisfy the **central** **limit** **theorem** (see condition (H) below). Then we give some spectral **and** mixing conditions that imply condition (H).

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(iii) T n S n x = x **and** T m S n x = S n−m x if n > m.
Then there exists a T -invariant strongly mixing (Gaussian) Borel probability measure µ on X with full support.
**Theorem** 1.2 has also been obtained in [MAP13] in a completely different way. The measure µ constructed in [MAP13] is not a Gaussian measure; in [MAP13], very few properties of this measure are proved. For instance, it is not known whether the norm k · k or the linear functionals hx ∗ , ·i belong to L 2 (X, µ), even if we can always ensure these properties, see Section 3. On the other hand, the dynamical system (T, µ) is conjugated to an easy strongly mixing dynamical system: a Bernoulli shift.

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argument, we can also recover asymptotic properties for continuous time quadratic variations, which may be used when dealing with increments of non linear functionals of Y instead of increments of Y . Let us come back to the theoretical part of this paper, which constitutes its core. We revisit Breuer Major’s **Theorem**, which is our main tool to obtain **Central** **Limit** Theorems, **and** use the powerful theory developed by Nourdin, Nualart, Ortiz-Latorre, Peccati, Tudor **and** others to do so. This is described in the next section **and** we refer to it for more details. We would like to attract attention to a remark, which has its own interest: under appropriate additional assumptions, the Malliavin

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that the consistency **and** the asymptotic normality of the robust estimator of the covariance have been discussed in [16]).
In the related context of the estimation of the fractal dimension of locally self-similar Gaussian processes [10] has proposed a robust estimator of the Hurst coefficient; instead of using the variance of the generalized discrete variations of the process (which are closely related to the wavelet coefficients, despite the facts that the motivations are quite different), this author proposes to use the empirical quantiles **and** the trimmed-means. The consistency **and** asymptotic normality of this estimator is established for a class of locally self-similar processes, using a Bahadur-type representation of the sample quantile; see also [9]. [28] proposes to replace the classical regression of the wavelet coefficients by a robust regression approach, based on Huberized M-estimators.

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kg 1 k ∞ ≤ ∞, kg 2 k ∞ ≤ ∞, Lip g 1 < ∞ **and** Lip g 2 < ∞.
A lot of usual time series are η-weakly dependent. Different examples of such time series will be studied in the following section: strong mixing processes (see Doukhan **and** Louhichi, 1999), GARCH(p, q) or ARCH(∞) processes (see Doukhan et al., 2004), causal or non causal linear processes (see Doukhan **and** Lang, 2002), causal or non causal bilinear processes (see Doukhan et al., 2005) **and** causal or non causal Volterra processes (see Doukhan, 2003). Now, we can specify the different assumptions used in the general functional **central** **limit** **theorem**:

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The aim of this work is to demonstrate that the steady-state analysis that has been used to derive FBA from ODEs is also completely within the reach of the probabilistic methods. We derive the consequences of applying the same simpli- fying assumption (that is, the system as the system has reached a steady-state in which the quantities of internal chemical species are constant) to the SSA. The consequence is the existence of a multivariate **central** **limit** **theorem** (CLT) for the trajectories where the limiting distribution is specified by the stoichiometry matrix **and** reaction probabilities which are the analogous of FBA fluxes. Thus our ap- proach needs as much information as FBA, that is to say mainly the stoichiometry, but is inherently stochastic. In the article, we derive the CLT for the stochastic trajectories of a reaction network in steady-state. Then, we present multiple theo- retical **and** practical applications of this result. For instance, we derive confidence regions for the aforementioned model validation problem (Figure 1 ) **and** we propose a constraints-based approach, similar to FBA, to integrate experimental data.

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