On the strong attraction of functions on dependent variables to the normal law

For symmetric functions on random variables from stationary sequences satisfying the uniformly strong mixing condition, the general conditions of attraction to the normal law in terms of distributions of individual items are obtained. The main result of the paper generalizes all known to present results of this type.

On the asymptotic normality of some sums of dependent random variables

A theorem on the asymptotic normality of the sum of dependent random variables is stated and proved. Conditions of the theorem are formulated in terms of a dependency graph which characterizes the relationships between random variables. This theorem is used to prove the asymptotic normality of the sum of functions defined on subsets of elements of the stationary sequence satisfying the strong mixing condition. As an illustration of possible applications of these theorems we give a theorem on the asymptotic normality of the number of empty cells if the random sequence of cells occupied by particles is a stationary sequence satisfying the uniform strong mixing condition.

PROBABILITY DISTRIBUTIONS OF SCATTERED SIGNALS IN A RANDOM MULTILAYER

This is a review paper on the study of the randomly scattered signals in a random multilayer based upon a stochastic and asymptotic formulation under strong mixing condition. This formulation generalizes the dominant Ito's formulation. The existence of a turning point of the random wave requires several type stochastic differential equations and the relevant limit theorems. The probability distributions of the randomly scattered signals have been obtained in the form of the Kolmogorov PDEs along the line of Khasminskii's limit theorem. This article demonstrates the step-by-step development of the relevant generators which contain the ultimate information for the probability distributions of the random signals.

A nearly independent, but non-strong mixing, triangular array

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.

A nearly independent, but non-strong mixing, triangular array

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.