Asymptotics for the survival probability in a killed branching random walk

Details Asymptotics for the survival probability in a killed branching random walk Asymptotics for the survival probability in a killed branching random walk

2008-11-03

arxiv.org/abs/0811.0262v3

Mathematics

Probability

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Scientific paper

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $\epsilon\to 0$, the probability in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where $\beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0

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