Moments of the first descending epoch for a random walk with negative drift

Sergey Foss, Timofei Prasolov

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We consider the first descending ladder epoch τ=min{n≥1:S n≤0} of a random walk S n=∑ 1 nξ i,n≥1 with i.d.d. summands having a negative drift Eξ=−a<0. Let ξ +=max(0,ξ 1). It is well-known that, for any α>1, the finiteness of E(ξ +) α implies the finiteness of Eτ α and, for any λ>0, the finiteness of Eexp(λξ +) implies that of Eexp(cτ) where c>0 is, in general, another constant that depends on the distribution of ξ 1. We consider the intermediate case, assuming that Eexp(g(ξ +))<∞ for a positive increasing function g such that lim inf x→∞g(x)/logx=∞ and lim sup x→∞g(x)/x=0, and that Eexp(λξ +)=∞, for all λ>0. Assuming a few further technical assumptions, we show that then Eexp((1−ɛ)g((1−δ)aτ))<∞, for any ɛ,δ∈(0,1).

Original languageEnglish
Article number109547
JournalStatistics and Probability Letters
Early online date1 Jun 2022
Publication statusPublished - Oct 2022


  • Descending ladder epoch
  • Existence of moments
  • Heavy tail
  • Negative drift
  • Random walk

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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