Annealed deviations of random walk in random scenery

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...

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...

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...

Quenched, Annealed and Functional Large Deviations for One-Dimensional Random Walk in Random Environment

Suppose that the integers are assigned random variables f! i g (taking values in the unit interval), which serve as an environment. This environment deenes a random walk fX n g (called a RWRE) which, when at i, moves one step to the right with probability ! i , and one step to the left with probability 1 ? ! i. When the f! i g sequence is i.i.d., Greven and den Hollander (1994) proved a large d...

Large Deviations for Random Walk in Random