Under a well-known scaling, supercritical Galton–Watson processes Z converge to a non-degenerate non-negative random limit variable W. We are dealing with the left tail (i.e. close to the origin) asymptotics of its law. In the Böttcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the Böttcher case, we improve a recent lower deviation probability result by describing the precise asymptotics under a logarithmic scaling (Theorem 7). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 8).
@article{1234469978,
author = {Fleischmann, Klaus and Wachtel, Vitali},
title = {On the left tail asymptotics for the limit law of supercritical Galton--Watson processes in the B\"ottcher case},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {45},
number = {1},
year = {2009},
pages = { 201-225},
language = {en},
url = {http://dml.mathdoc.fr/item/1234469978}
}
Fleischmann, Klaus; Wachtel, Vitali. On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case. Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 201-225. http://gdmltest.u-ga.fr/item/1234469978/