Martingale approach to subexponential asymptotics for random walks

Abstract

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 $\tau =\min \{n\ge 1:S_n\le 0\}$.

9 pages