Martingale approach to subexponential asymptotics for random walks

Consider the random walk S n = ξ 1+,..., + ξ n with independent and identically distributed increments and negative mean Eξ = - m <0. Let M = sup 0 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 P(M τ > x), where M τ = max 0<i<τ S i and τ = min{n ≥ 1: S n ≤ 0}.