Chover-type laws of the iterated logarithm for Kesten-Spitzer random walks in random sceneries belonging to the domain of stable attraction
Summary: Let \(X = \{X_i, i \geq 1 \}\) be a sequence of real valued random variables, \(S_0 = 0\) and \(S_k = \sum_{i = 1}^k X_i(k \geq 1)\). Let \(\sigma = \{\sigma(x), x \in \mathbb{Z} \}\) be a sequence of real valued random variables which are independent of \(X\)'s. Denote by \(K_n = \sum_{k = 0}^n \sigma(\lfloor S_k \rfloor)(n \geq 0)\) Kesten-Spitzer random walk in random scenery, where \(\lfloor a \rfloor\) means the unique integer satisfying \(\lfloor a \rfloor \leq a < \lfloor a \rfloor + 1\). 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.
scientific article; zbMATH DE number 7026717
scientific article