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

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).