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, Tome 45 (2009) no. 1, pp. 201-225.

Par un changement d'échelle bien connu, on obtient que les processus de Galton-Watson supercritiques sur Z convergent vers une variable aléatoire non-degénerée W. Nous considérons les estimées asymptotiques à gauche (près de l'origine) de la distribution. Dans le cas Böttcher (quand il y a au moins deux progénitures en chaque point), nous obtenons l'asymptotique exacte présentant un comportement oscillatoire (Théorème 1). Sous une autre hypothèse raisonnable, les oscillations s'annulent (Corollaire 2). Pour le cas Böttcher, nous présentons un résultat sur la probabilité des grandes déviations, amélioré en exprimant l'asymptotique exacte sous un scaling logarithmique (Théorème 7). En imposant d'autres conditions, nous obtenons des asymptotiques plus raffinées (Théorème 8), c'est-à-dire sans log-scaling.

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).

DOI : 10.1214/07-AIHP162
Classification : 60J80, 60F10
Mots-clés : lower deviation probabilities, Schröder case, Böttcher case, logarithmic asymptotics, fine asymptotics, precise asymptotics, oscillations 
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Fleischmann, Klaus; 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, Tome 45 (2009) no. 1, pp. 201-225. doi : 10.1214/07-AIHP162. http://www.numdam.org/articles/10.1214/07-AIHP162/

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