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

top
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

top

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

top
  1. [1] M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79 (1988) 543–623. Zbl0635.60090MR966175
  2. [2] J. D. Biggins. The growth of iterates of multivariate generating functions. Trans. Amer. Math. Soc. 360 (2008) 4305–4334. Zbl1158.39017MR2395174
  3. [3] J. D. Biggins and N. H. Bingham. Near-constancy phenomena in branching processes. Math. Proc. Cambridge Philos. Soc. 110 (1991) 545–558. Zbl0749.60077MR1120488
  4. [4] J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. Appl. Probab. 25 (1993) 757–772. Zbl0796.60090MR1241927
  5. [5] N. H. Bingham. Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 (1976) 217–242. Zbl0338.60051MR410961
  6. [6] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25A (1988) 215–228. Zbl0669.60078MR974583
  7. [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. Zbl0297.60044MR362525
  8. [8] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, 1987. Zbl0617.26001MR898871
  9. [9] M. S. Dubuc. La densite de la loi-limite d’un processus en cascade expansif. Z. Wahrsch. Verw. Gebiete 19 (1971) 281–290. Zbl0215.25603MR300353
  10. [10] W. Feller. An Introduction to Probability Theory and Its Applications, volume II, 2nd edition. Wiley, New York, 1971. Zbl0219.60003MR270403
  11. [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. Zbl0566.30023MR757658
  12. [12] K. Fleischmann and V. Wachtel. Lower deviation probabilities for supercritical Galton–Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 233–255. Zbl1112.60066MR2303121
  13. [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. Zbl0819.60076MR1316808
  14. [14] T. E. Harris. Branching processes. Ann. Math. Statist. 19 (1948) 474–494. Zbl0041.45603MR27465
  15. [15] O. D. Jones. Multivariate Böttcher equation for polynomials with nonnegative coefficients. Aequationes Math. 63 (2002) 251–265. Zbl1001.39027MR1904719
  16. [16] O. D. Jones. Large deviations for supercritical multitype branching processes. J. Appl. Probab. 41 (2004) 703–720. Zbl1075.60110MR2074818

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.