On the exact asymptotic behaviour of the distribution of ladder epochs

Let T+ denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E(X)=0,E(X2)=[infinity], and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, PT+[greater-or-equal, slanted]n~n1/[beta]-1L+(n) as n-->[infinity], with 1<[beta]<2 and L+ slowly varying, if and only ifP{X[greater-or-equal, slanted]x}~ 1/{x[beta]L(x)} as x-->+[infinity], with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L+ and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.