The sequential empirical process of a random walk in random scenery

Abstract

A random walk in random scenery (Yn) n∈N is given by Yn = ξ Sn for a random walk (Sn) n∈N and iid random variables (ξn) n∈Z . In this paper, we will show the weak convergence of the sequential empirical process, i.e. the centered and rescaled empirical distribution function. The limit process shows a new type of behavior, combining properties of the limit in the independent case (roughness of the paths) and in the long range dependent case (selfsimilarity).

Empirical processes for recurrent and transient random walks in random scenery

“…The problem we investigate in the present paper has already been studied in [35] in the case where (S n ) n∈N is a recurrent random walk in Z such that (n − 1 α S n ) n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1,2].…”

“…is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n∈N is a random walk evolving in Z d , independent of the ξ's. In [35], the case where (Sn)n∈N is a recurrent random walk in Z such that (n − 1 α Sn) n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n∈N is either :…”

“…

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s, t)) s,t∈[0,1] with Wn(s, t) := ⌊nt⌋ k=1 1 {ξ S k ≤s} − s where (ξx, x ∈ Z d ) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n∈N is a random walk evolving in Z d , independent of the ξ's. In [35], the case where (Sn)n∈N is a recurrent random walk in Z such that (n − 1 α Sn) n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n∈N is either :(a) a transient random walk in Z d , (b) a recurrent random walk in Z d such that (n − 1 d Sn) n≥1 converges in distribution to a stable distribution of index d ∈ {1, 2}.

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Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

“…Finally for any arbitrary transient random walk, it can be shown that the sequence { −1/2 , ∈ N} is asymptotically normal (see for instance Spitzer [7] page 53). Among others, we can cite strong approximation results [8][9][10], laws of the iterated logarithm [11][12][13], limit theorems for correlated sceneries or walks [14][15][16][17], large and moderate deviations results [18][19][20][21][22], and ergodic and mixing properties (see the survey [23]). …”