Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Abstract

Let = { , ≥ 1} be a sequence of real valued random variables, 0 = 0 and = ∑ =1 ( ≥ 1). Let = { ( ), ∈ Z} be a sequence of real valued random variables which are independent of 's. Denote by = ∑ =0 (⌊ ⌋) ( ≥ 0) Kesten-Spitzer random walk in random scenery, where ⌊ ⌋ means the unique integer satisfying ⌊ ⌋ ≤ < ⌊ ⌋ + 1. It is assumed that 's belong to the domain of attraction of a stable law with index 0 < < 2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery . The obtained results supplement to some corresponding results in the literature.