Martingale approach to subexponential asymptotics for random walks

2011/11/29 by Denis Denisov, Denisov, Denis, Vitali Wachtel +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60G70 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60G70

paper · pdf · doi:10.48550/arxiv.1111.6810

9 pages

arxiv created 2011/11/29 · openalex publication_date 2011/11/29 · arxiv updated 2011/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the random walk S_n=ξ₁+...+ξ_n with independent and identically distributed increments and negative mean \mathbf Eξ=-m<0. Let M=sup_{0≤ i} S_i be the supremum of the random walk. In this note we present derivation of asymptotics for \mathbf P(M>x), x→∞ 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_τ>x), where M_τ=max_{0≤ i<τ}S_i and τ=min\n≥ 1: S_n≤ 0 \.

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