On extremes of a random walk with positive drift over an intermediate regularly varying time interval

2026/07/25 by Sergey Foss, Dmitry Korshunov
Mathematics · #math.PR

paper · pdf

Abstract

We consider a random walk \S_n\ with a finite positive drift that is stopped at a random time τ having an intermediate regularly varying distribution. We assume that the jump distribution is lighter-tailed than the distribution of τ. Under these conditions, we show that the tails of the distributions of S_τ and M_τ = max_{k≤ τ} S_k are asymptotically equivalent and are determined by the tail of τ, while the random walk \S_n\ contributes only through the law of large numbers.

Citations

• Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes
• Tails of random sums of a heavy-tailed number of light-tailed terms

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