Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:34:08.987Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  09 December 2016

Alexander Iksanov  and
Zakhar Kabluchko
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Zakhar Kabluchko*
Affiliation:
Westfälische Wilhelms-Universität Münster
*
* Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
** Postal address:Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (W n (θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (W n (2θ))n∈ℕ0 is uniformly integrable and that var W 1(θ) is finite, we prove a functional central limit theorem for the tail process (W ∞(θ)-W n+r (θ))r∈ℕ0 and a law of the iterated logarithm for W ∞(θ)-W n (θ) as n→∞.

Keywords

The Biggins martingalebranching random walkcentral limit theoremlaw of the iterated logarithm

MSC classification

Primary: 60G42: Martingales with discrete parameter
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1178 - 1192
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adamczak, R. and Miłoś, P. (2015).CLT for Ornstei‒Uhlenbeck branching particle system.Electron. J. Prob. 20,42.CrossRefGoogle Scholar
[2] Aizenman, M.,Lebowitz, J. L. and Ruelle, D. (1987).Some rigorous results on the Sherrington‒Kirkpatrick spin glass model.Commun. Math. Phys. 112,3–20.(Correction:116(1988), S27.)CrossRefGoogle Scholar
[3] Alsmeyer, G. and Iksanov, A. (2009).A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks.Electron. J. Prob. 14,289–313.Google Scholar
[4] Alsmeyer, G.,Iksanov, A.,Polotskiy, S. and Rösler, U. (2009).Exponential rate of $L p -convergence of intrinsic martingales in supercritical branching random walks.Theory Stoch. Process. 15,1–18.Google Scholar
[5] Asmussen, S. and Hering, H. (1983).Branching Processes(Progress Prob. Statist. 3).Birkhäuser,Boston, MA.CrossRefGoogle Scholar
[6] Asmussen, S. and Keiding, N. (1978).Martingale central limit theorems and asymptotic estimation theory for multitype branching processes.Adv. Appl. Prob. 10,109–129.CrossRefGoogle Scholar
[7] Athreya, K. B. (1968).Some results on multitype continuous time Markov branching processes.Ann. Math. Statist. 39,347–357.Google Scholar
[8] Athreya, K. B. and Ney, P. E. (2004).Branching Processes.Dover,Mineola, NY.Google Scholar
[9] Biggins, J. D. (1977).Martingale convergence in the branching random walk.J. Appl. Prob. 14,25–37.Google Scholar
[10] Biggins, J. D. (1998).Lindley-type equations in the branching random walk.Stoch. Process. Appl. 75,105–133.Google Scholar
[11] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1989).Regular Variation (Encyclopedia Math. Appl. 27).Cambridge University Press.Google Scholar
[12] Bovier, A.,Kurkova, I. and Löwe, M. (2002).Fluctuations of the free energy in the REM and the p-spin SK models.Ann. Prob. 30,605–651.Google Scholar
[13] Bühler, W. J. (1969).Ein zentraler Grenzwertsatz für Verzweigungsprozesse.Z. Wahrscheinlichkeitsth. 11,139–141.Google Scholar
[14] Grübel, R. and Kabluchko, Z. (2016).A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees.To appear in Ann. Appl. Prob. Google Scholar
[15] Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Application.Academic Press,New York.Google Scholar
[16] Heyde, C. C. (1970).A rate of convergence result for the super-critical Galton‒Watson process.J. Appl. Prob. 7,451–454.Google Scholar
[17] Heyde, C. C. (1971).Some central limit analogues for supercritical Galton‒Watson processes.J. Appl. Prob. 8,52–59.Google Scholar
[18] Heyde, C. C. and Brown, B. M. (1971).An invariance principle and some convergence rate results for branching processes.Z. Wahrscheinlichkeitsth. 20,271–278.CrossRefGoogle Scholar
[19] Heyde, C. C. and Leslie, J. R. (1971).Improved classical limit analogues for Galton‒Watson processes with or without immigration.Bull. Austral. Math. Soc. 5,145–155.CrossRefGoogle Scholar
[20] Iksanov, O. M. (2006).On the rate of convergence of a regular martingale related to a branching random walk.Ukrainian Math. J. 58,368–387.CrossRefGoogle Scholar
[21] Iksanov, A. and Meiners, M. (2010).Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks.J. Appl. Prob. 47,513–525.Google Scholar
[22] Kabluchko, Z. and Klimovsky, A. (2014).Generalized random energy model at complex temperatures.Preprint. Available at https://arxiv.org/abs/1402.2142.Google Scholar
[23] Kesten, H. and Stigum, B. P. (1966).Additional limit theorems for indecomposable multidimensional Galton‒Watson processes.Ann. Math. Statist. 37,1463–1481.Google Scholar
[24] Lyons, R. (1997).A simple path to Biggins' martingale convergence for branching random walk.In Classical and Modern Branching Processes,Springer,New York,pp. 217–221.Google Scholar
[25] Neininger, R. (2015).Refined Quicksort asymptotics.Random Structures Algorithms 46,346–361.Google Scholar
[26] Ren, Y.-X.,Song, R. and Zhang, R. (2015).Central limit theorems for supercritical superprocesses.Stoch. Process. Appl. 125,428–457.Google Scholar
[27] Rösler, U.,Topchii, V. A. and Vatutin, V. A. (2002).The rate of convergence for weighted branching processes.Siberian Adv. Math. 12,57–82.Google Scholar
[28] Sulzbach, H. (2016).On martingale tail sums for the path length in random trees. To appear in Random Structures Algorithms.Google Scholar