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

We define a stochastic process X={Xn,n=0,1,2,.} in terms of cumulative sums of the sequence K1,K2,. of integer-valued random variables in such a way that if the Ki are independent, identically distributed and nonnegative, then X is a Bienayme-Galton-Watson branching process. By exploiting the fact that 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.