Operator Approach to Weak Convergence of Measures and Limit Theorems for Random Operators
Yu. N. Orlov, В. Ж. Сакбаев, E. V. Shmidt
Abstract
Abstract The generalized weak convergence of a sequence of measures is induced by the convergence of the linear operators generated by the measures. A corresponding generalization of the notion of convergence over a distribution is introduced. Generalized convergence over the distribution of a sequence of compositions of independent random transformations is investigated. The connection between limit distributions and semigroups that solve initial-boundary value problems for evolution equations is established. In the case of a sequence of compositions of independent random transformations of the shift to a random vector of Euclidean space, the results obtained coincide with the central limit theorem for sums of independent random vectors.