Asymptotic results for empirical measures of weighted sums of independent random variables

We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric matrices, the speed of the almost sure convergence is described by the normal approximation and by the large deviation principle

Limiting operations for sequences of quantum random variables and a convergence theorem for quantum martingales

We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.Comment: 11 pages, 0 figure

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein's method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed

An information-theoretic Central Limit Theorem for finitely susceptible FKG systems

We adapt arguments concerning entropy-theoretic convergence from the independent case to the case of FKG random variables. FKG systems are chosen since their dependence structure is controlled through covariance alone, though in the sequel we use many of the same arguments for weakly dependent random variables. As in previous work of Barron and Johnson, we consider random variables perturbed by small normals, since the FKG property gives us control of the resulting densities. We need to impose a finite susceptibility condition -- that is, the covariance between one random variable and the sum of all the random variables should remain finite.Comment: 17 page

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics on the Poisson space

Almost sure convergence for weighted sums of pairwise PQD random variables

We obtain Marcinkiewicz-Zygmund strong laws of large numbers for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable $X \in \mathscr{L}_{p}$, $1 \leqslant p < 2$. We use our results to establish the strong consistency of estimators which emerge from regression models having pairwise positively quadrant dependent errors.Comment: 20 page