This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.

A k × n array with entries from an "alphabet" A = { 0, 1, ..., q - 1 } of size q is said to form a t-covering array (resp. orthogonal array) if each t × n submatrix of the array contains, among its columns, at least one (resp. exactly one) occurrence of each t-letter word from A (we must thus have n = qt for an orthogonal array to exist and n ≥ qt for a t -covering array). In this paper, we continue the agenda laid down in Godbole et al. (2009) in which the notion of consecutive covering arrays was defined and motivated; a detailed study of these arrays for the special case q = 2, has also carried out by the same authors. In the present article we use first a Markov chain embedding method to exhibit, for general values of q, the probability distribution function of the random variable W = Wk, n, t defined as the number of sets of t consecutive rows for which the submatrix in question is missing at least one word. We then use the Chen-Stein method (Arratia et al., 1989, 1990) to provide upper bounds on the total variation error incurred while approximating L (W) by a Poisson distribution Po (λ) with the same mean as W. Last but not least, the Poisson approximation is used as the basis of a new statistical test to detect run-based discrepancies in an array of q-ary data.

In this thesis, abstract bounds for the normal approximation of Poisson functionals are computed by the Malliavin-Stein method and used to derive central limit theorems for problems from stochastic geometry. As a Poisson functional we denote a random variable depending on a Poisson point process. It is known from stochastic analysis that every square integrable Poisson functional has a representation as a (possibly infinite) sum of multiple Wiener-Ito integrals. This decomposition is called Wiener-Itô chaos expansion, and the integrands are denoted as kernels of the Wiener-Itô chaos expansion. An explicit formula for these kernels is known due to Last and Penrose. Via their Wiener-Itô chaos expansions the so-called Malliavin operators are defined. By combining Malliavin calculus and Stein's method, a well-known technique to derive limit theorems in probability theory, bounds for the normal approximation of Poisson functionals in the Wasserstein distance and vectors of Poisson functionals in a similar distance were obtained by Peccati, Sole, Taqqu, and Utzet and Peccati and Zheng, respectively. An analogous bound for the univariate normal approximation in Kolmogorov distance is derived. In order to evaluate these bounds, one has to compute the expectation of products of multiple Wiener-Itô integrals, which are complicated sums of deterministic integrals. Therefore, the bounds for the normal approximation of Poisson functionals reduce to sums of integrals depending on the kernels of the Wiener-Itô chaos expansion. The strategy to derive central limit theorems for Poisson functionals is to compute the kernels of their Wiener-Itô chaos expansions, to put the kernels in the bounds for the normal approximation, and to show that the bounds vanish asymptotically. By this approach, central limit theorems for some problems from stochastic geometry are derived. Univariate and multivariate central limit theorems for some functionals of the intersection process of Poisson k-flats and the number of vertices and the total edge length of a Gilbert graph are shown. These Poisson functionals are so-called Poisson U-statistics which have an easier structure since their Wiener-Itô chaos expansions are finite, i.e. their Wiener-Itô chaos expansions consist of finitely many multiple Wiener-Itô integrals. As examples for Poisson functionals with infinite Wiener-Itô chaos expansions, central limit theorems for the volume of the Poisson-Voronoi approximation of a convex set and the intrinsic volumes of Boolean models are proven.

L'objet de cette thèse est l'étude de certaines propriétés analytiques et asymptotiques des processus de Markov, et de leurs applications à la méthode de Stein. Le point de vue considéré consiste à déployer des inégalités fonctionnelles pour majorer la distance entre lois de probabilité. La première partie porte sur l'étude asymptotique de processus de Markov inhomogènes en temps via des inégalités de type Poincaré, établies par l'analyse spectrale fine de l'opérateur de transition. On se place d'abord dans le cadre du théorème central limite, qui affirme que la somme renormalisée de variables aléatoires converge vers la mesure gaussienne, et l'étude est consacrée à l'obtention d'une borne à la Berry-Esseen permettant de quantifier cette convergence. La distance choisie est une quantité naturelle et encore non étudiée dans ce cadre, la distance du chi-2, complétant ainsi la littérature relative à d'autres distances (Kolmogorov, variation totale, Wasserstein). Toujours dans le contexte non-homogène, on s'intéresse ensuite à un processus peu mélangeant relié à un algorithme stochastique de recherche de médiane. Ce processus évolue par sauts de deux types (droite ou gauche), dont la taille et l'intensité dépendent du temps. Une majoration de la distance de Wasserstein d'ordre 1 entre la loi du processus et la mesure gaussienne est établie dans le cas où celle-ci est invariante sous la dynamique considérée, et étendue à des exemples où seule la normalité asymptotique est vérifiée. La seconde partie s'attache à l'étude des entrelacements entre processus de Markov (homogènes) et gradients, qu'on peut interpréter comme un raffinement du critère de Bakry-Emery, et leur application à la méthode de Stein, qui est un ensemble de techniques permettant de majorer la distance entre deux mesures de probabilité. On prouve l'existence de relations d'entrelacement du second ordre pour les processus de naissance-mort, allant ainsi plus loin que les relations du premier ordre connues. Ces relations sont mises à profit pour construire une méthode originale et universelle d'évaluation des facteurs de Stein relatifs aux mesures de probabilité discrètes, qui forment une composante essentielle de la méthode de Stein-Chen. / The object of this thesis is the study of some analytical and asymptotic properties of Markov processes, and their applications to Stein's method. The point of view consists in the development of functional inequalities in order to obtain upper-bounds on the distance between probability distributions. The first part is devoted to the asymptotic study of time-inhomogeneous Markov processes through Poincaré-like inequalities, established by precise estimates on the spectrum of the transition operator. The first investigation takes place within the framework of the Central Limit Theorem, which states the convergence of the renormalized sum of random variables towards the normal distribution. It results in the statement of a Berry-Esseen bound allowing to quantify this convergence with respect to the chi-2 distance, a natural quantity which had not been investigated in this setting. It therefore extends similar results relative to other distances (Kolmogorov, total variation, Wasserstein). Keeping with the non-homogeneous framework, we consider a weakly mixing process linked to a stochastic algorithm for median approximation. This process evolves by jumps of two sorts (to the right or to the left) with time-dependent size and intensity. An upper-bound on the Wasserstein distance of order 1 between the marginal distribution of the process and the normal distribution is provided when the latter is invariant under the dynamic, and extended to examples where only the asymptotic normality stands. The second part concerns intertwining relations between (homogeneous) Markov processes and gradients, which can be seen as refinment of the Bakry-Emery criterion, and their application to Stein's method, a collection of techniques to estimate the distance between two probability distributions. Second order intertwinings for birth-death processes are stated, going one step further than the existing first order relations. These relations are then exploited to construct an original and universal method of evaluation of discrete Stein's factors, a key component of Stein-Chen's method.