On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case

Klaus Fleischmann; Vitali Wachtel

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 201-225
  • ISSN: 0246-0203

Abstract

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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).

How to cite

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Fleischmann, Klaus, and Wachtel, Vitali. "On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 201-225. <http://eudml.org/doc/78016>.

@article{Fleischmann2009,
abstract = {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).},
author = {Fleischmann, Klaus, Wachtel, Vitali},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {lower deviation probabilities; Schröder case; Böttcher case; logarithmic asymptotics; fine asymptotics; precise asymptotics; oscillations},
language = {eng},
number = {1},
pages = {201-225},
publisher = {Gauthier-Villars},
title = {On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case},
url = {http://eudml.org/doc/78016},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Fleischmann, Klaus
AU - Wachtel, Vitali
TI - On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 201
EP - 225
AB - 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).
LA - eng
KW - lower deviation probabilities; Schröder case; Böttcher case; logarithmic asymptotics; fine asymptotics; precise asymptotics; oscillations
UR - http://eudml.org/doc/78016
ER -

References

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