## Abstract

We consider the first descending ladder epoch τ=min{n≥1:S _{n}≤0} of a random walk S _{n}=∑ _{1} ^{n}ξ _{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 language | English |
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Article number | 109547 |

Journal | Statistics and Probability Letters |

Volume | 189 |

Early online date | 1 Jun 2022 |

DOIs | |

Publication status | Published - Oct 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty