Generalisations of the Bienayme-Galton-Watson Branching Process via its Representation as an Embedded Random Walk

Abstract

We define a stochastic process $\mathscr{X} = \{X_n, n = 0,1,2, \ldots\}$ in terms of cumulative sums of the sequence $K_1, K_2, \ldots$ of integer-valued random variables in such a way that if the $K_i$ are independent, identically distributed and nonnegative, then $\mathscr{X}$ is a Bienayme-Galton-Watson branching process. By exploiting the fact that $\mathscr{X}$ is in a sense embedded in a random walk, we show that some standard branching process results hold in more general settings. We also prove a new type of limit result.