TY - JOUR

T1 - Moments of the first descending epoch for a random walk with negative drift

AU - Foss, Sergey

AU - Prasolov, Timofei

N1 - Funding Information: The work is supported by Mathematical Center in Akademgorodok, Russia under agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © 2022 Elsevier B.V.

PY - 2022/10

Y1 - 2022/10

N2 - We consider the first descending ladder epoch τ=min{n≥1:Sn≤0} of a random walk Sn=∑1nξi,n≥1 with i.d.d. summands having a negative drift Eξ=−a<0. Let ξ+=max(0,ξ1). It is well-known that, for any α>1, the finiteness of E(ξ+)α implies the finiteness of Eτα 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 ξ1. We consider the intermediate case, assuming that Eexp(g(ξ+))<∞ for a positive increasing function g such that lim infx→∞g(x)/logx=∞ and lim supx→∞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).

AB - We consider the first descending ladder epoch τ=min{n≥1:Sn≤0} of a random walk Sn=∑1nξi,n≥1 with i.d.d. summands having a negative drift Eξ=−a<0. Let ξ+=max(0,ξ1). It is well-known that, for any α>1, the finiteness of E(ξ+)α implies the finiteness of Eτα 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 ξ1. We consider the intermediate case, assuming that Eexp(g(ξ+))<∞ for a positive increasing function g such that lim infx→∞g(x)/logx=∞ and lim supx→∞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).

KW - Descending ladder epoch

KW - Existence of moments

KW - Heavy tail

KW - Negative drift

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=85132405993&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/eb756b9d-d8d8-334e-a854-ad8bfd3ce5bc/

U2 - 10.1016/j.spl.2022.109547

DO - 10.1016/j.spl.2022.109547

M3 - Article

AN - SCOPUS:85132405993

VL - 189

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

M1 - 109547

ER -