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

Let X = {X-i, i >= 1) be a sequence of real valued random variables, S-o = 0 and S-k = Sigma i=1(k) X-i (k >= 1). Let sigma = {(x),x is an element of Z) be a sequence of real valued random variables which are independent of X's. Denote by K-n = Sigma(n)(k=0) sigma(left perpendicularS(k)rightperpendicular), (n >= 0) Kesten-Spitzer random walk in random scenery, where left perpendicular alpha rightperpendicular means the unique integer satisfying left perpendicular alpha rightperpendicular <= alpha < left perpendicular alpha rightperpendicular + 1. It is assumed that sigma's belong to the domain of attraction of a stable law with index 0 < beta < 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 K-n. The obtained results supplement to some corresponding results in the literature.