Gaussian approximations for random vectors

Abstract

We present several refinements on the fluctuations of sequences of random vectors (with values in the Euclidean space R d ) which converge after normalization to a multidimensional Gaussian distribution. More precisely we refine such results in two directions: first we give conditions under which one can obtain bounds on the speed of convergence to the multidimensional Gaussian distribution, and then we provide a setting in which one can obtain precise moderate or large deviations (in particular we see at which scale the Gaussian approximation for the tails ceases to hold and how the symmetry of the Gaussian tails is then broken). These results extend some of our earlier works obtained for real valued random variables, but they are not simple extensions, as some new phenomena are observed that could not be visible in one dimension. Even for very simple objects such as the symmetric random walk in Z d , we observe a loss of symmetry that we can quantify for walks conditioned to be far away from the origin. Also, unlike the one dimensional case where the Kolmogorov distance is natural, in the multidimensional case there is no more such a canonical distance. We choose to work with the so-called convex distance, and as a consequence the geometry of the Borel measurable sets that we consider shall play an important role (also making the proofs more complicated). We illustrate our results with some examples such as correlated random walks, the characteristic polynomials of random unitary matrices, or pattern countings in random graphs. CONTENTS1. Introduction 1.1. Notational conventions 1.2. Mod-Gaussian convergence 1.3. The one-dimensional case 1.4. Main results and outline of the article Acknowledgement 2. Estimates on the speed of convergence 2.1. Convex distance between probability measures 2.2. Smoothing techniques 2.3. Distance between Fourier transforms 2.4. Berry-Esseen type estimates 3. Large deviation results 3.1. Sum of a Gaussian noise and an independent random variable 3.2. Asymptotics of the probabilities of cones 3.3. Approximation of spherical sectors by unions of cones 4. Examples of multi-dimensional convergent sequences 4.1. First examples 4.2. The method of cumulants 4.3. Sparse dependency graphs 4.4. Empirical measures of Markov chains References