The multidimensional analog of Yu. V. Linnik's theorem on decompositions of a composition of Gaussian and Poisson laws

If the sum of two independent $n$-dimensional random vectors is distributed according to the law which is a composition of $n$-dimensional Gaussian and $n$-dimensional Poisson laws then each of these vectors is distributed according to the law which is a composition of $n$-dimensional Gaussian and $n$-dimensional Poisson laws.