Limit of the environment viewed from Sina\"i's walk

TL;DR

This paper studies Sina"i's walk, showing the convergence of the environment viewed from the particle to a limit measure, and derives ergodic theorems, LLN, and CLT for additive functionals of the walk.

Contribution

It explicitly characterizes the limit measure for the environment viewed from Sina"i's walk and extends ergodic theorems and limit laws to recurrent cases.

Findings

Convergence of the empirical environment measure to a limit measure.

Establishment of an ergodic theorem in law for additive functionals.

Proof of LLN and CLT for sums of functions of the walk's steps.

Abstract

For Sina\"i's walk (X_k) we show that the empirical measure of the environment seen from the particle (\bar\w_k) converges in law to some random measure S. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of (\bar\w_k) . When the limit in this "in law" ergodic theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums sum_{k=1}^nf(\Delta X_k), where f is bounded and depending on the steps \Delta X_k:=X_{k+1}-X_k.