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 ≤ 0 } of a random walk ∑ ξ i , with i.d.d. summands having negative drift E − < . Let + max ( ) It is well-known that, for any α > finiteness implies and, λ exp that c where is, in general, another constant depends on distribution intermediate case, assuming g ∞ positive increasing function such lim inf x → / log and sup all Assuming few further technical assumptions, we show then ɛ δ ∈