On the characterization of multidimensional normal law by the independence of linear statistics

Let $\{X_j\}$ be a sequence of independent random vectors in $R^k$ and $\{A_j,B_j\}$ be a sequence of pairs of nonsingular real $(k\times k)$-matrices. It is shown that every $X_j$ has $k$-dimensional normal distribution if linear statistics (1) converge with probability 1 to independent random vectors and the condition (2) is satisfied.