Limit theorem for random walk in weakly dependent random scenery

Abstract

Let $S=(S_k{)}_{k\ge 0}$ be a random walk on $\mathbb{Z}$ and $\xi =({\xi }_i{)}_{i\in \mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $({\Sigma }_n{)}_{n\ge 0}$ where $${\Sigma }_n=\sum _{k=0}^n{\xi }_{S_k},n\in \mathbb{N}.$$ Under a weak dependence assumption on the scenery $\xi$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).

References in corpus (2)

• Discrete approximation of a stable self-similar stationary increments process
• Central limit theorem for sampled sums of dependent random variables