Empirical processes for recurrent and transient random walks in random scenery

Abstract

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s, t)) s,t∈ [0,1] withis 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}.

Limit theorems for additive functionals of random walks in random scenery

“…Limit theorems have been extended by Bolthausen [6] (for the case α = β = 2 for random walks of dimension d = 2), by Deligiannidis and Utev [19] (for the case α = d ∈ {1, 2}, β = 2, providing some correction to [6]) and by Castell, Guillotin-Plantard and the author [12] (when α ≤ d and β < 2), completing the picture for the convergence in the sense of distribution and for the functional limit theorem (except in the case α ≤ 1 and β = 1 for which the tightness remains an open question). Since the seminal works by Borodin and by Kesten and Spitzer, random walks in random scenery and the Kesten and Spitzer process ∆ have been the object of various studies (let us mention for example [33,50,29,3,27,25,28,2]).…”

Limit theorems for additive functionals of random walks in random scenery

“…Limit theorems have been extended by Bolthausen [6] (for the case α = β = 2 for random walks of dimension d = 2), by Deligiannidis and Utev [19] (α = d ∈ {1, 2}, β = 2, providing some correction to [6]) and by Castell, Guillotin-Plantard and the author [12] (when α ≤ d and β < 2), completing the picture for the convergence in the sense of distribution and for the functional limit theorem (except in the case α ≤ 1 and β = 1). Since the seminal works by Borodin and by Kesten and Spitzer, random walks in random scenery and the Kesten and Spitzer process ∆ have been the object of various studies (let us mention for example [33,50,29,3,27,25,28,2]).…”