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Almost sure central limit theorem for branching random walks in random environment. (English) Zbl 1210.60108
Summary: We consider the branching random walks in \(d\)-dimensional integer lattice with time-space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When \(d\geq 3\) and the fluctuation of environment satisfies a certain uniform square integrability. then it is nondegenerate, and we prove a central limit theorem for the density of the population in terms of almost sure convergence.

MSC:
60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
60F15 Strong limit theorems
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