Strong law of large numbers for random walks in weakly dependent random scenery

TL;DR

The paper addresses the strong law of large numbers for random walks in random scenery when the scenery is non-stationary and only weakly dependent. It develops a framework based on local and intersection local times, with $Z_n=\sum_i N_n(i)\xi_i$, and proves SLLN-type results under $\theta_2$-weak dependence of the scenery and varying regimes of walk dependence, including long-range dependence. The authors extend prior stationary/independent results to non-stationary settings and, in a related case, establish SLLN with $Z_n/n^{\tau}\to E\xi_0$ for $\tau>3/4$ under $\theta_1$-weak dependence; these contributions broaden the applicability of SLLN for random walks in random scenery. The results provide a rigorous understanding of when the cumulative scenery observed along a random walk converges almost surely, with implications for models where the environment evolves with limited dependence.

Abstract

In this brief note, we study the strong law of large numbers for random walks in random scenery. Under the assumptions that the random scenery is non-stationary and satisfies weakly dependent condition with an appropriate rate, we establish strong law of large numbers for random walks in random scenery. Our results extend the known results in the literature.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Proposition 3
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 7