On convergence of distributions for sums of independent random vectors with randomly change of components

We derive new results on convergence of distributions for sums of independent random vectors with randomly changed components in scheme of series. In particular, a multidimensional central limit theorem is proved. If the random change of components is defined by a Poisson process then we arrive at results on convergence of finitely dimension distributions of psi-processes. In Gaussian case, the limit process is the Ornstein–Uhlenbeck process. We discuss a replacement of the Poisson process by processes with non-negative integer increments.