A Criterion of Boundedness of Discrete Branching Random Walk

A general model of a branching random walk in Z is considered, where the branching and displacements occur with probabilities determined by the position of a parent. A necessary and sufficient condition is given for the random variable $$ M = \sup \;\max \;\;{X_{n,k}}\;\;\;n \geqslant 0,1 \leqslant k \leqslant {N_n} $$ (1) to be finite. Here X n,k is the position of the k-th offspring in the n-th generation. The condition is stated in terms of a naturally arising linear functional equation. A number of examples are discussed, where the condition may be verified.