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, Volume 45 (2009) no. 1, p. 201-225

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).

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.

DOI : https://doi.org/10.1214/07-AIHP162
Classification:  60J80,  60F10
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {201-225},
     doi = {10.1214/07-AIHP162},
     zbl = {1175.60075},
     mrnumber = {2500235},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 45 (2009) no. 1, pp. 201-225. doi : 10.1214/07-AIHP162. http://www.numdam.org/item/AIHPB_2009__45_1_201_0/

[1] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988) 543-623. | MR 966175 | Zbl 0635.60090

[2] J. D. Biggins. The growth of iterates of multivariate generating functions. Trans. Amer. Math. Soc. 360 (2008) 4305-4334. | MR 2395174 | Zbl 1158.39017

[3] J. D. Biggins and N. H. Bingham. Near-constancy phenomena in branching processes. Math. Proc. Cambridge Philos. Soc. 110 (1991) 545-558. | MR 1120488 | Zbl 0749.60077

[4] J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. Appl. Probab. 25 (1993) 757-772. | MR 1241927 | Zbl 0796.60090

[5] N. H. Bingham. Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 (1976) 217-242. | MR 410961 | Zbl 0338.60051

[6] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25A (1988) 215-228. | MR 974583 | Zbl 0669.60078

[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 362525 | Zbl 0297.60044

[8] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, 1987. | MR 898871 | Zbl 0617.26001

[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 300353 | Zbl 0215.25603

[10] W. Feller. An Introduction to Probability Theory and Its Applications, volume II, 2nd edition. Wiley, New York, 1971. | MR 270403 | Zbl 0219.60003

[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 757658 | Zbl 0566.30023

[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 2303121 | Zbl 1112.60066

[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 1316808 | Zbl 0819.60076

[14] T. E. Harris. Branching processes. Ann. Math. Statist. 19 (1948) 474-494. | MR 27465 | Zbl 0041.45603

[15] O. D. Jones. Multivariate Böttcher equation for polynomials with nonnegative coefficients. Aequationes Math. 63 (2002) 251-265. | MR 1904719 | Zbl 1001.39027

[16] O. D. Jones. Large deviations for supercritical multitype branching processes. J. Appl. Probab. 41 (2004) 703-720. | MR 2074818 | Zbl 1075.60110