Local limit theorems for linear generated random vectors

Let $\{\xi_k\}$ be a sequence of independent identically distributed vandom variables $a_{kn}\in\mathbb R^m$, $S_n=(S_n^1,\dots,S_n^m)=\sum_ka_{kn}\xi_k.$ Sufficient conditions are obtained for the convergence in variance of distributions of $S_n (n\to\infty)$, to the multivariate-normal distribution. Application of the results to the convergence in variance of distributions of functionels of random processes is given.