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

Let X i , 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 X i , n ≧ 1, and M n = max 0 ≦ k ≦ n S k . In the case where the X i are such that Σ 1 ∞ n −1 Pr( S n > 0) < ∞, we have lim n →∞ M n = M which is finite with probability one, while in the case where Σ 1 ∞ n −1 Pr( S n < 0) < ∞, a limit theorem for M n has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ 1 ∞ n −1 Pr( S n < 0) < ∞, Σ 1 ∞ n −1 Pr( S n > 0) < ∞ (the case of oscillation of the random walk generated by the S n ) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the X i themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for M n in the case of oscillation.