Local limit theory and large deviations for supercritical Branching processes

Abstract

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Z_n:n≥1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n↗∞, and use this to study conditional large deviations of {Y_Zn:n≥1}, where Y_n satisfies an LDP, particularly of {Z_n⁻¹Z_n+1:n≥1} conditioned on Z_n≥v_n.