Local limit theorems for linearly generated random vectors

Abstract

Let {ξk} be a sequence of independent, identically distributed random variables, $$a_{kn} \in R^m ,S_n = \left( {S_n^1 , ..., S_n^m } \right) = \mathop \Sigma \limits_k a_{kn} \xi _{k.} $$ . One obtains sufficient conditions for the convergence in variation of the distributions of Sn, n→∞, to the multidimensional normal distribution. One also gives an application of these results to the convergence in variation of distributions of functionals of random processes.