Moment Asymptotics for Multitype Branching Random Walks in Random Environment

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• 作者：Onur Gün ; Wolfgang K?nig ; Ozren Sekulovi?
• 关键词：Multitype branching random walk ; Feynman–Kac ; type formula ; Variational analysis ; Annealed moments ; Large deviations ; 60J80 ; 60J55 ; 60F10 ; 60K37 ; 60J10
• 刊名：Journal of Theoretical Probability
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：28
• 期：4
• 页码：1726-1742
• 全文大小：477 KB
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• 作者单位：Onur Gün (1)
  Wolfgang K?nig (1) (2)
  Ozren Sekulovi? (3)
  
  1. Weierstrass Institute Berlin, Mohrenstr. 39, 10117?, Berlin, Germany
  2. Institute for Mathematics, TU Berlin, Str.?des 17.?Juni 136, 10623?, Berlin, Germany
  3. University of Montenegro, Cetinjska 2, 81 000?, Podgorica, Montenegro
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

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 \(1/\rho _{ij}\) for the upper tail of the mean number of \(j\) type particles produced by an \(i\) 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