On the left tail asymptotics for the limit law of supercritical Galton--Watson processes in the Böttcher case
- Fleischmann, Klaus
- Wachtel, Vitali
2010 Mathematics Subject Classification
- 60J80 60F10
- Lower deviation probabilities, Schröder case, Böttcher case, logarithmic asymptotics, fine asymptotics, precise asymptotics, tiny oscillations
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. lose to the origin) asymptotics of its law. In the Bötcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing tiny 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 3). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 4).