Generalisations of the Bienaymé-Galton-Watson branching process via its representations as an embedded random walk (Q1345591)

Let \(\{K_ n\}\) be a sequence of integer-valued random variables and define sequences \(\{X_ n\}\), \(\{T_ n\}\) by \(X_ 0 = 1\), \(X_ 1 = K_ 1\), \(T_ n = \sum^ n_{j = 0} X_ j\) and \(X_{n + 1} = \left( \sum^{T_{n-1} + X_ n}_{j = T_{n - 1} + 1} K_ j \right) I(X_ n \geq 1)\), \(n \geq 1\). If the random variables \(\{K_ n\}\) are i.i.d. and non-negative, then \(\{X_ n\}\) is a Bienaymé-Galton-Watson branching process and \(\{T_ n\}\) is its total-progeny process. Generalizations of branching-process results are obtained when the non- negativity and the i.i.d. assumptions on the \(\{K_ n\}\) are relaxed. The paper considers (i) the probability of extinction, (ii) the almost- sure asymptotic behaviour of \(\{X_ n\}\) when the probability of extinction is less than 1 and (iii) the rate of asymptotic convergence.