Martingale approach to subexponential asymptotics for random walks

Details Martingale approach to subexponential asymptotics for random walks Martingale approach to subexponential asymptotics for random walks

2011-11-29

arxiv.org/abs/1111.6810v1

Mathematics

Probability

9 pages

Scientific paper

Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and $\tau=\min\{n\ge 1: S_n\le 0 \}$.

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