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Martingale approach to subexponential asymptotics for random walks

   
@article{ECP1757,
	author = {Denis Denisov and Vitali Wachtel},
	title = {Martingale approach to subexponential asymptotics for random walks},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {random walk; supremum; cycle maximum; heavy-tailed  distribution; stopping time},
	abstract = {
Consider the random walk $S_n=\xi_1+\cdots+\xi_n$ 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 $\tau=\min\{n\ge 1: S_n\le 0 \}$.
},
	pages = {no. 6, 1-9},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-1757},    
        url = {http://ecp.ejpecp.org/article/view/1757}}