On one limit theorem for branching random walks with a finite number of particle types

We consider a branching random walk on the lattice $\mathbb{Z}^d$, $d\in \mathbb{N}$, in which at any point of $\mathbb{Z}^d$ a particle of every type can die or produce an arbitrary number of offsprings of different types. The walk of a particle of each type on $\mathbb{Z}^d$ is described by a symmetric homogeneous and irreducible random walk. We assume that the branching intensity of particles of any type at a point $x\in \mathbb{Z}^d$ tends to zero as $\|x\|\to\infty$, and an additional condition is fulfilled on the parameters of the branching random walk, guaranteeing exponential in time growth of the mean number of particles of each type at each point $\mathbb{Z}^d$. Under these assumptions we prove the limit theorem on the convergence of normalised number of particles of each type at an arbitrary fixed point $y_{0}\in \mathbb{Z}^d$ as $t\rightarrow\infty$ to the limit in mean square. The proof is based on an approximation of the normalised number of particles by some non-negative martingale.