# and central limit theorem.

## Top PDF and central limit theorem.: ### Mixing properties and central limit theorem for associated point processes

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.
En savoir plus ### Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

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  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.
En savoir plus ### Rates in the Central Limit Theorem and diffusion approximation via Stein's Method

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
En savoir plus ### Dependent Lindeberg central limit theorem and some applications

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.
En savoir plus ### A central limit theorem for adaptive and interacting Markov chains

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.
En savoir plus ### Wiener integrals with respect to the Hermite process and a Non-Central Limit Theorem

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  and . 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 ### A central limit theorem for the exodic tree and exodic matching in geometrical probability

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] ### Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem

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 . 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.
En savoir plus ### Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions

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
En savoir plus ### Central Limit Theorem and Large Deviation Principle for Continuous Time Open Quantum Walks

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 . 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  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  for link between discrete and continuous time models in the context of OQW, one can also consult  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]).
En savoir plus ### Asymptotics and Scalings for Large Closed Product-Form Networks via the Central Limit Theorem

and both tend to infinity. This approach has been considered by Knessl and Tier , Kogan and Birman [6, 7, 1] and Malyshev and Yakovlev . However, it relies on purely analytical tools, which are difficult to use in a more general setting and, in our opinion, do not really give a structural explanation of the phenomena involved. ### Central Limit Theorem and bootstrap procedure for Wasserstein's variations with application to structural relationships between distributions

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.
En savoir plus ### A Central Limit Theorem for the Length of the Longest Common Subsequence in Random Words

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
En savoir plus ### Rates of convergence for minimal distances in the central limit theorem under projective criteria

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. ### A note on the central limit theorem for two-fold stochastic random walks in a random environment

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. , Theorem 3, p. 121). However the statement and the argument used in  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).
En savoir plus ### Central limit theorems in linear dynamics

(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.
En savoir plus ### Central Limit Theorems and Quadratic Variations in terms of Spectral Density

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
En savoir plus ### Central limit theorem for the robust log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context

that the consistency and the asymptotic normality of the robust estimator of the covariance have been discussed in ). In the related context of the estimation of the fractal dimension of locally self-similar Gaussian processes  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 .  proposes to replace the classical regression of the wavelet coefficients by a robust regression approach, based on Huberized M-estimators.
En savoir plus ### A functional limit theorem for η -weakly dependent processes and its applications

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:
En savoir plus ### Multivariate Normal Approximation for the Stochastic Simulation Algorithm: limit theorem and applications

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.
En savoir plus