On the strong law of large numbers for random walks in random scenery

Let $\left(\Omega,\mathcal{F},\mathbf{P} \right)$ be a probability space. The random variables we deal with are all defined on $\left(\Omega,\mathcal{F},\mathbf{P} \right)$. Let $\left(\xi_{i} \right)_{i\in \mathbf{Z}}$ and $\left(X_{n} \right)_{n\in \mathbf{N}}$ be two independent sequences of independent identically distributed (i.i.d.) random variables taking values in $\mathbf{R}$ and $\mathbf{Z}$, respectively. The sequence $\left(\xi_{i} \right)_{i\in \mathbf{Z}}$ is called the random scenery. The sequence $\left(X_{n} \right)_{n\in \mathbf{N}}$ is the sequence of increments of the random walk $\left(S_{n} \right)_{n\geq 0}$ defined by $S_{0}=0$ and $S_{n}:=X_{1}+\dots +X_{n}$ for $n\in \mathbf{N}$.
We define the random walk in random scenery as the process $\left(Z_{n}\right)_{n\geq 0}$ given by
$$ Z_{0}=0, \ \ Z_{n}=\sum_{k=0}^{n}\xi_{S_{k}}, \ \ n\in \mathbf{N}. $$
There have been substantial research works devoted to studying the asymptotic behavior of $Z_{n}$. The process $Z_{n}$ was introduced by Borodin [1] and independently by Kesten and Spitzer [2]. In the talk, we shall discuss the validity of the strong law of large numbers for $Z_{n}$ when the random scenery $(\xi_{i})_{i \in \mathbf{Z}}$ is assumed to be either independent or weakly dependent.