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

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).