On the maximum of sums of random variables and the supremum functional for stable processes

Abstract

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S 0= 0, S n = Σ i=1 n Xi , n ≧ 1, and Mn = max0 ≦ k ≦ n Sk . In the case where the Xi are such that Σ1 ∞ n −1Pr(Sn > 0) < ∞, we have lim n→∞M n = M which is finite with probability one, while in the case where Σ1 ∞ n −1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1 ∞ n −1Pr(Sn < 0) < ∞, Σ1 ∞ n −1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn ) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.