Conditional Limit Laws and Inference for Generation Sizes of Branching Processes

Abstract

Let {Z_n:n≥0} denote a single type supercritical branching process initiated by a single ancestor. This paper studies the asymptotic behavior of the history of generation sizes conditioned on different notions of information about the “current” population size. A “suppression property” under the large deviation conditioning, namely that R_n≡Z_n+1/Z_n>a, is observed. Furthermore, under a more refined conditioning, the asymptotic aposteriori distribution of the original offspring distribution is developed. Implications of our results to conditional consistency property of age is discussed.