Large deviations for the maximum of a branching random walk with\n stretched exponential tails

2020/06/14 by Piotr Dyszewski, Dyszewski, Piotr, Nina Gantert +3 · 1 citation
Mathematics · Decision Sciences · #Stochastic processes and statistical mechanics #Probability and Risk Models #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2006.09207

Abstract

We prove large deviation results for the position of the rightmost particle,\ndenoted by M_n, in a one-dimensional branching random walk in a case when\nCram 'er's condition is not satisfied. More precisely we consider step size\ndistributions with stretched exponential upper and lower tails, i.e.~both tails\ndecay as e^-|t|r for some r\∈( 0,1). It is known that in this case,\nM_n grows as n^1/r and in particular faster than linearly in n. Our\nmain result is a large deviation principle for the laws of n^-1/rM_n . In\nthe proof we use a comparison with the maximum of (a random number of)\nindependent random walks, denoted by M_n, and we show a large\ndeviation principle for the laws of n^-1/r M_n as well.\n

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• Asymptotic behavior of the extremal position in a multi-type branching random walk with heavy-tailed displacements

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