In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

Let (Sn)n≥0 be a $\mathbb Z$ -random walk and $(\xi_{x})_{x\in \mathbb Z}$ be a sequence of independent andidentically distributed $\mathbb R$ -valued random variables,independent of the random walk. Let h be a measurable, symmetricfunction defined on $\mathbb R^2$ with values in $\mathbb R$ . We study theweak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$ , withvalues in D[0,1] the set of right continuous real-valuedfunctions with left limits, defined by \[ \sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].\] Statistical applications are presented, in particular we prove a strong law of large numbersfor U-statistics indexed by a one-dimensional random walk using a result of [1].

Let (Sk)k≥0 be a Markov chain with state space E and (ξx)x∊E be a family of ℝp-valued random vectors assumed independent of the Markov chain. The ξx could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

We find the limiting distribution of , where {Bu}u≧0 is the standard Brownian motion on ℝd, V is a particular random potential and {an}n≧1 is a normalizing sequence.