Limit theorems for low dimensional generalized $(T,T^{-1})$ transformations

Abstract

We consider generalized $(T, T^{-1})$ transformations such that the base map satisfies a multiple mixing local limit theorem and anticoncentration large deviation bounds and in the fiber we have $\mathbb{R}^d$ actions with $d=1$ or $2$ which are exponentially mixing of all orders. If the skewing cocycle has zero drift, we show that the ergodic sums satisfy the same limit theorems as the random walks in random scenery studied by Kesten and Spitzer (1979) and Bolthausen (1989). The proofs rely on the quenched CLT for the fiber action and the control of the quenched variance. This paper complements our previous work where the classical central limit theorem is obtained for a large class of generalized $(T, T^{-1})$ transformations.