Moderate deviations for the self-normalized random walk in random scenery

Moderate Deviations for Random Walk in Random Scenery

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. In the case d ≥ 4 we eve...

Annealed Deviations of Random Walk in Random Scenery

Let (Zn)n∈N be a d-dimensional random walk in random scenery, i.e., Zn = ∑n−1 k=0 Y (Sk) with (Sk)k∈N0 a random walk in Z d and (Y (z))z∈Zd an i.i.d. scenery, independent of the walk. The walker’s steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P( 1 nZn > bn) for various choices of sequences (bn)n in [1,∞). Depending on (b...

Large Deviations for Random Walk in Random

Deviations of a Random Walk in a Random Scenery with Stretched Exponential Tails

Let (Zn)n∈N0 be a d-dimensional random walk in random scenery, i.e., Zn = ∑n−1 k=0 YSk with (Sk)k∈N0 a random walk in Z d and (Yz)z∈Zd an i.i.d. scenery, independent of the walk. We assume that the random variables Yz have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of P( 1 nZn > tn) for all se...

From charged polymers to random walk in random scenery

We prove that two seemingly-different models of random walk in random environment are generically quite close to one another. One model comes from statistical physics, and describes the behavior of a randomlycharged random polymer. The other model comes from probability theory, and was originally designed to describe a large family of asymptotically self-similar processes that have stationary i...