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

作者：Uchiyama Kohei^*

来源：Stochastic Processes and Their Applications, 2011, 121(9): 1938-1961.

DOI：10.1016/j.spa.2011.04.009

• Ladder height
• Ladder epoch
• Potential function
• Spitzer's condition

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.