A note on random walk in random scenery

Abstract

We consider a random walk in random scenery { X n = η ( S 0 ) + ⋯ + η ( S n ) , n ∈ N } , where a centered walk { S n , n ∈ N } is independent of the scenery { η ( x ) , x ∈ Z d } , consisting of symmetric i.i.d. with tail distribution P ( η ( x ) > t ) ∼ exp ( − c α t α ) , with 1 ⩽ α < d / 2 . We study the probability, when averaged over both randomness, that { X n > n y } for y > 0 , and n large. In this note, we show that the large deviation estimate is of order exp ( − c ( n y ) a ) , with a = α / ( α + 1 ) .