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

Published online by Cambridge University Press:  14 July 2016

Pál Révész
Affiliation:
Technische Universität Wien
Jay Rosen
Affiliation:
City University of New York
Zhan Shi
Affiliation:
Université Paris VI
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Abstract

Given an ℝ d -valued supercritical branching Wiener process, let ψ(A, T) be the number of particles in A ⊂ ℝ d 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.

Type
Research Papers
Copyright
© Applied Probability Trust 2005 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137151.CrossRefGoogle Scholar
Chen, X. (2001). Exact convergence rates for the distribution of particles in branching random walks. Ann. Appl. Prob. 11, 12421262.Google Scholar
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.CrossRefGoogle 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. 2531.Google Scholar

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