Limit distributions arising in branching random walks on integer lattices

Abstract

The paper completes the investigation of limit distribution of the number of particles at the source of branching in the model of critical catalytic branching random walk on \( {{\mathbb Z}^d}\;d \in {\mathbb N} \). Limit theorems of such kind were established only for d = 1, 2, 3, 4 under the assumption that, at the initial moment, there is a single particle at the source of branching. We prove their analog for \( d \geqslant 5 \). Moreover, in any dimension, we generalize the previous results by permitting the initial particle to start at an arbitrary point of the lattice.