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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Uchiyama, Kohei

关键词: Ladder height;    Ladder epoch;    Potential function;    Spitzer's condition;   

DOI  :  10.1016/j.spa.2011.04.009

来源: Elsevier

【 摘 要 】

This paper concerns a recurrent random walk on the real line R 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] similar to -x(-alpha)/Gamma(1 - alpha)L(x) (x -> infinity) if and only if P[T > n] similar to n(-1+1/alpha) /Gamma(a alpha)L alpha* (n), where L alpha* 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. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free