Complete left tail asymptotic for the density of branching processes in the Schröder case
Consider a Galton-Watson process with offspring distribution satisfying \(p_0=0\) and \(p_1\in (0, 1)\). Assume that the mean number of offspring is finite.The author uses Fourier analysis method to study the density \(p(x)\) of the limit of the fundamental martingale and establishes a complete expansion of \(p(x)\). This sharpens the result obtained by \textit{J. D. Biggins} and \textit{N. H. Bingham} [Adv. Appl. Probab. 25, No. 4, 757--772 (1993; Zbl 0796.60090)].
Schröder and Poincaré-type functional equations
scientific article; zbMATH DE number 7882013
scientific article