Abstract
Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<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 Kn. The obtained results supplement to some corresponding results in the literature.
Highlights
Let X = {Xi, i ≥ 1} be a sequence of real valued random variables, S0 = 0 and Sk = ∑ki=1 Xi (k ≥ 1)
{σ(x), x ∈ Z} be 0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋ ≤ a < ⌊a⌋ + 1
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 Kn
Summary
Kesten and Spitzer [1] proved that when the random walk and the random scenery belong to the domains of attraction of different stable laws of indices 1
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