Moment Asymptotics for Multitype Branching Random Walks in Random Environment

Onur Gün (), Wolfgang König () and Ozren Sekulović ()
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Onur Gün: Weierstrass Institute Berlin
Wolfgang König: Weierstrass Institute Berlin
Ozren Sekulović: University of Montenegro

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1726-1742

Abstract: Abstract We study a discrete-time multitype branching random walk on a finite space with finite set of types. Particles move in space according to a Markov chain whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman–Kac formula. We choose Weibull-type distributions with parameter $$1/\rho _{ij}$$ 1 / ρ i j for the upper tail of the mean number of $$j$$ j type particles produced by an $$i$$ i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Keywords: Multitype branching random walk; Feynman–Kac-type formula; Variational analysis; Annealed moments; Large deviations; 60J80; 60J55; 60F10; 60K37; 60J10 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-014-0551-2

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