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On the lattice case of an almost-sure renewal theorem for branching random walks

Part of: Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  19 February 2016

Dimitris Gatzouras
Dimitris Gatzouras*
Affiliation:
University of Cambridge
*
∗ Postal address: Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece. Email address: gatzoura@math.uoc.gr
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Abstract

We formulate and verify an almost-sure lattice renewal theorem for branching random walks, whose non-lattice analogue is originally due to Nerman. We also identify the limit in these renewal theorems (both lattice and non-lattice) as the limit of Kingman's well-known martingale multiplied by a deterministic factor.

Keywords

Branching random walkalmost-sure renewal theoremlattice case

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 720 - 737
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Biggins, J. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.Google Scholar
[2] Doney, R. (1972). A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
[3] Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[4] Gatzouras, D. (2000). Lacunarity of self-similar and stochastically self-similar sets. Trans. Amer. Math. Soc. 352, 1953–1983.CrossRefGoogle Scholar
[5] Kingman, J. F. C. (1975). The first birth problem for an age dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
[6] Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes, eds Athreya, K. and Jagers, P., IMA Vol. Math. Appl. 84, Springer, New York, pp. 217–221.Google Scholar
[7] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
[8] Meyer, P. A. (1972). Martingales and Stochastic Integrals I. Springer, Berlin.Google Scholar
[9] Nerman, O. (1981). On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.CrossRefGoogle Scholar
[10] Shiryaev, A.N. (1996). Probability, 2nd edn. Springer, New York.Google Scholar