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

Part of: Stochastic processes Markov 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.
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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 

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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