Local Limit Theory and Large Deviations for Supercritical Branching Processes

In this paper we study several aspects of the growth of a supercritical Galton-Watson process$\lbrace Z_n:n \geq 1\rbrace$, and bring out some criticality phenomena determined by the$Schr\ddot{o}der$constant. We develop the local limit theory of Zn, that is, the behavior of$P(Z_n = \upsilon_n)$as$\upsilon_n \nearrow \infty$, and use this to study conditional large deviations of$\lbrace Y_{Z_n} :n \geq 1\rbrace$, where Ynsatisfies an LDP, particularly of$\lbrace Z_n^{-1} Z_{n+1} :n \geq 1\rbrace$conditioned on$Z_n \geq \upsilon_n$.