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
@article{AIHPB_2009__45_1_201_0,
     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 = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {201--225},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP162},
     mrnumber = {2500235},
     zbl = {1175.60075},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/07-AIHP162/}
}
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
SP  - 201
EP  - 225
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/07-AIHP162/
DO  - 10.1214/07-AIHP162
LA  - en
ID  - AIHPB_2009__45_1_201_0
ER  - 
%0 Journal Article
%A Fleischmann, Klaus
%A Wachtel, Vitali
%T On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 201-225
%V 45
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/07-AIHP162/
%R 10.1214/07-AIHP162
%G en
%F AIHPB_2009__45_1_201_0
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://archive.numdam.org/articles/10.1214/07-AIHP162/

[1] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988) 543-623. | MR | Zbl

[2] J. D. Biggins. The growth of iterates of multivariate generating functions. Trans. Amer. Math. Soc. 360 (2008) 4305-4334. | MR | Zbl

[3] J. D. Biggins and N. H. Bingham. Near-constancy phenomena in branching processes. Math. Proc. Cambridge Philos. Soc. 110 (1991) 545-558. | MR | Zbl

[4] J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. Appl. Probab. 25 (1993) 757-772. | MR | Zbl

[5] N. H. Bingham. Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 (1976) 217-242. | MR | Zbl

[6] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25A (1988) 215-228. | MR | Zbl

[7] N. H. Bingham and R. A. Doney. Asymptotic properties of supercritical branching processes. I. The Galton-Watson processes. Adv. Appl. Probab. 6 (1974) 711-731. | MR | Zbl

[8] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, 1987. | MR | Zbl

[9] M. S. Dubuc. La densite de la loi-limite d'un processus en cascade expansif. Z. Wahrsch. Verw. Gebiete 19 (1971) 281-290. | MR | Zbl

[10] W. Feller. An Introduction to Probability Theory and Its Applications, volume II, 2nd edition. Wiley, New York, 1971. | MR | Zbl

[11] P. Flajolet and A. M. Odlyzko. Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations. Math. Proc. Cambridge Philos. Soc. 96 (1984) 237-253. | MR | Zbl

[12] K. Fleischmann and V. Wachtel. Lower deviation probabilities for supercritical Galton-Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 233-255. | Numdam | MR | Zbl

[13] B. M. Hambly. On constant tail behaviour for the limiting random variable in a supercritical branching process. J. Appl. Probab. 32 (1995) 267-273. | MR | Zbl

[14] T. E. Harris. Branching processes. Ann. Math. Statist. 19 (1948) 474-494. | MR | Zbl

[15] O. D. Jones. Multivariate Böttcher equation for polynomials with nonnegative coefficients. Aequationes Math. 63 (2002) 251-265. | MR | Zbl

[16] O. D. Jones. Large deviations for supercritical multitype branching processes. J. Appl. Probab. 41 (2004) 703-720. | MR | Zbl

Cité par Sources :