Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),. . . be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + . . . + X(n), n = 1, 2,. . ., and define the Markov stopping time ηy = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,. . . . In the case E{pipe}X{pipe}3 < ∞, the following relation was obtained in [8]: as n → ∞, where the constant R and the bounded sequence vn were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence, for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit for every fixed y ≥ 0 under the condition EX2 < ∞ only; In [1], an explicit form of the limit was found under the same condition EX2 < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture. © 2010 Allerton Press, Inc.