Moderate deviations for a random walk in random scenery

Abstract

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions d ≥ 2, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d ≥ 4 we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d ≥ 3, an important ingredient in the proofs are new concentration inequalities for selfintersection local times of random walks, which are of independent interest, whilst for d = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

Upper tails for intersection local times of random walks in supercritical dimensions

“…[MS93], or in the analysis of stochastic processes in random environments, see e.g. [HKM06,GKS07,AC07,FMW08]. In the latter models dependence between a moving particle and a random environment frequently comes from the particle's ability to revisit sites with a (in some sense) attractive environment, and therefore measures of self-intersection quantify the degree of dependence between movement and environment.…”

A quenched functional central limit theorem for planar random walks in random sceneries

“…The case of a transient random walk (i.e., a < d) has also been treated in [10] (see also [29,23,7]): rescaling by n 1/b one obtains as limit a stable process of index b. Other results on RWRS include strong approximation results and laws of the iterated logarithm [14,15,24], limit theorems for correlated sceneries or walks [13,22], large and moderate deviations results [1,9,12,19,20], ergodic and mixing properties [17].…”

Moderate Deviations for Stable Random Walks in Random Scenery

“…They proved that if {ξ k , k ≥ 0} and {ζ x , x ∈ Z} belong to domains of attraction of different stable laws with indices 1 < α ≤ 2 and 0 < β ≤ 2, respectively, then n −δ X n converges in distribution as n → ∞ to a nondegenerate variable, where δ = 1 − 1/α + 1/αβ. More generally, if the underlying random walks have finite variance, Gantert Moderate deviations for stable random walks 281 et al [11] analyzed the deviations P(X n /n > b n ) for various choices of sequences {b n } n∈N in [1, ∞) with b n → ∞ as n → ∞ in the case of arbitrary sceneries unbounded to +∞ and Fleischmann et al [10] proved moderate deviation principles for X n in the d ≥ 2 case with the random sceneries satisfying Cramér's condition. For example, when S n is a simple random walk in Z d , Csáki et al [8] studied the strong invariance principle for X n in the d = 2 case, Asselah and Castell [2] estimated the probability that X n is large in the d ≥ 5 case, and Asselah [1] investigated the moderate deviation for X n in the d = 3 case.…”