F. Götze

Asymptotic expansions for sums of weakly dependent random vectors

SummaryIt is shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors. The error of approximation has ordero(n−(s−2)/2) if(i)the moments of orders+1 are uniformly bounded(ii)a conditional Cramér-condition holds(iii)the random vectors can be approximated by other random vectors which satisfy a strong mixing condition and a Markov type condition. The strong mixing coefficients in (iii) are decreasing at an exponential rate. The above conditions can easily be checked and are often satisfied when the sequence of random vectors is a Gaussian, or a Markov, or an autoregressive process. Explicit formulas are given for the distribution of finite Fourier transforms of a strictly stationary time series.

THE CIRCULAR LAW FOR RANDOM MATRICES

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending results to a larger class of sparse matrices.

The arithmetic of distributions in free probability theory

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

Limit Theorems for Spectra of Random Matrices with Martingale Structure

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semicircle law.

The rate of convergence for spectra of GUE and LUE matrix ensembles

We obtain optimal bounds of order O(n−1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

A Berry-Esséen bound for student's statistic in the non-I.I.D. case

We establish a Berry-Esséen bound for Students statistic for independent (nonidentically) distributed random variables. In particular, the bound implies a sharp estimate similar to the classical Berry-Esséen bound. In the i.i.d. case it yields sufficient conditions for the Central Limit Theorem for studentized sums. For non-i.i.d. random variables the bound shows that the Lindeberg condition is sufficient for the Central Limit Theorem for studentized sums.

Rate of convergence and edgeworth-type expansion in the entropic central limit theorem

An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.

Limit theorems in free probability theory. I

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle \( \mathbb{T} \) we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

On Gaussian and Bernoulli covariance representations

We discuss several applications, to large deviations for smooth functions of Gaussian random vectors, of a covariance representation in Gauss space. The existence of this type of representation characterizes Gaussian measures. New representations for Bernoulli measures are also derived, recovering some known inequalities.

The Accuracy of Gaussian Approximation in Banach Spaces

Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.