Publication date
March 1982
DOI Publisher
Published by Elsevier B.V. ISSN
0304-4149 Citation count (estimate)
12Abstract
AbstractLet 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)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, PT+⩾n∽n1β−1L+(n) as n→∞, with 1<β<2 and L+ slowly varying, if and only ifP{X⩾x}∽ 1/{xβL(x)} as x→+∞, 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
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