Existence and positivity of the limit in processes with a branching structure

Abstract

We define a stochastic process {X n } based on partial sums of a sequence of integer-valued random variables (K 0,K 1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K 1,K 2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {c n } with c n+1/c n → a as n → ∞ such that X n /c n converges almost surely to a finite random variable which is positive on the event {X n ↛ 0}. The result is extended to the case of exchangeable summands.