Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Klaus Fleischmann and Vitali Wachtel

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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).

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 201-225.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469978

Digital Object Identifier
doi:10.1214/07-AIHP162

Mathematical Reviews number (MathSciNet)
MR2500235

Zentralblatt MATH identifier
1175.60075

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

Keywords
Lower deviation probabilities Schröder case Böttcher case Logarithmic asymptotics Fine asymptotics Precise asymptotics Oscillations

Citation

Fleischmann, Klaus; Wachtel, Vitali. On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 201--225. doi:10.1214/07-AIHP162. https://projecteuclid.org/euclid.aihp/1234469978


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References

  • [1] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988) 543–623.
  • [2] J. D. Biggins. The growth of iterates of multivariate generating functions. Trans. Amer. Math. Soc. 360 (2008) 4305–4334.
  • [3] J. D. Biggins and N. H. Bingham. Near-constancy phenomena in branching processes. Math. Proc. Cambridge Philos. Soc. 110 (1991) 545–558.
  • [4] J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. Appl. Probab. 25 (1993) 757–772.
  • [5] N. H. Bingham. Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 (1976) 217–242.
  • [6] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25A (1988) 215–228.
  • [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.
  • [8] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, 1987.
  • [9] M. S. Dubuc. La densite de la loi-limite d’un processus en cascade expansif. Z. Wahrsch. Verw. Gebiete 19 (1971) 281–290.
  • [10] W. Feller. An Introduction to Probability Theory and Its Applications, volume II, 2nd edition. Wiley, New York, 1971.
  • [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.
  • [12] K. Fleischmann and V. Wachtel. Lower deviation probabilities for supercritical Galton–Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 233–255.
  • [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.
  • [14] T. E. Harris. Branching processes. Ann. Math. Statist. 19 (1948) 474–494.
  • [15] O. D. Jones. Multivariate Böttcher equation for polynomials with nonnegative coefficients. Aequationes Math. 63 (2002) 251–265.
  • [16] O. D. Jones. Large deviations for supercritical multitype branching processes. J. Appl. Probab. 41 (2004) 703–720.