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Large-time asymptotics for the density of a branching Wiener process

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Pál Révész ,
Jay Rosen  and
Zhan Shi
Pál Révész*
Affiliation:
Technische Universität Wien
Jay Rosen*
Affiliation:
City University of New York
Zhan Shi*
Affiliation:
Université Paris VI
*
∗Postal address: Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107, A-1040 Vienna, Austria. Email address: [email protected]
∗∗Postal address: Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, New York, NY 10314, USA. Email address: [email protected]
∗∗∗Postal address: Laboratoire de Probabilités UMR 7599, Université Paris VI, 4 place Jussieu, F-75252 Paris Cedex 05, France. Email address: [email protected]
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Abstract

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Given an ℝd-valued supercritical branching Wiener process, let ψ(A, T) be the number of particles in A ⊂ ℝd at time T (T = 0, 1, 2, …). We provide a complete asymptotic expansion of ψ(A, T) as T → ∞, generalizing the work of X. Chen.

Keywords

Branching Wiener processdistribution of particles

MSC classification

Primary: 60F15: Strong theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1081 - 1094
Copyright
© Applied Probability Trust 2005 

References

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Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Révész, P. (1994). Random Walks of Infinitely Many Particles. World Scientific, Singapore.Google Scholar
Révész, P. (2004). A prediction problem of the branching random walk. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), Applied Probability Trust, Sheffield, pp. 25–31.Google Scholar