A note on summability of ladder heights and the distributions of ladder epochs for random walks

This paper concerns a recurrent random walk on the real line and obtains a purely analytic result concerning the characteristic function, which is useful for dealing with some problems of probabilistic interest for the walk of infinite variance: it reduces them to the case when the increment variable X takes only values from {...,-2,-1,0,1}. Under the finite expectation of ascending ladder height of the walk, it is shown that given a constant 1<[alpha]<2 and a slowly varying function L(x) at infinity, P[X<-x]~-x-[alpha]/[Gamma](1-[alpha])L(x) (x-->[infinity]) if and only if , where is a de Bruijn [alpha]-conjugate of L and T denotes the first epoch when the walk hits (-[infinity],0]. Analogous results are obtained in the cases [alpha]=1 or 2. The method also provides another derivation of Chow's integrability criterion for the expectation of the ladder height to be finite.