The extremes of random walks in random sceneries

Abstract

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.

Extremes for transient random walks in random sceneries under weak independence conditions

“…The main idea to derive Theorem 1 is to adapt [8] to our context. We think that our method combined with Kallenberg's theorem ensures that the point process of exceedances converges to a Poisson point process, in the same spirit as Theorem 3 in [3]. More precisely, if the threshold is of the form u n = u n (x) = a n x + b n , for some x ∈ R, and if we let τ k = inf{m ∈ N + , #{S 1 , .…”

“…More recently, Franke and Saigo [2,3] have investigated extremes for a sequence of dependent random variables which do not satisfy the conditions D(u n ) and D ′ (u n ). More precisely, they consider the following problem.…”

“…Now, let {ξ(k), k ∈ Z} be a family of R-valued i.i.d random variables independent of the sequence {X k , k ∈ N + }. In the sense of [3], the sequence {ξ(S n ), n ∈ N + } is called a random walk in a random scenery. Franke and Saigo derive limit theorems for the maximum of the first n terms of {ξ(S n ), n ∈ N + } as n goes to infinity.…”

“…Franke and Saigo derive limit theorems for the maximum of the first n terms of {ξ(S n ), n ∈ N + } as n goes to infinity. An adaptation of Theorem 1 in [3] shows that in the transient case, i.e. α < 1, the following property holds: if nP (ξ(0) > u n ) −→ n→∞ τ for some sequence (u n ) and for some τ > 0, then…”

The sequential empirical process of a random walk in random scenery

“…It converges weakly to a self-similar process with exponent b > 1 2 , which has smooth paths even if the random variables (ξ n ) n∈Z are in the domain of attraction of a Lévy process with jumps, see [14]. Other results include the law of the iterated logarithm (Khoshnevisan and Lewis [15]), large deviations (Gantert, König, and Shi [10]), extremes (Franke and Saigo [9]) and U -statistics (Guillotin-Plantard and Ladret [12], Franke, Pène, and Wendler [8]). As far as we know, there are no results on the empirical process of a random walk in random scenery.…”