Exceedance Problems And Binary Spliting In Branching Stochastic Processes

We explore the relationship between branching processes and random sums of indicators. As a tool for obtaining the\ud results, we look into a simple lemma about the distributional properties of a sequence of random variables with members falling in certain random interval. Then, we consider the number of families in Bienayme-Galton-Watson branching processes where size is within a random interval. We obtain limit theorems for subcritical, critical and supercritical processes with and without immigration