Strassen's invariance principle for random walk in random environment

TL;DR

This paper establishes a strong invariance principle for random walks in random environments on integer lattices, using martingale methods and fractional coboundaries, bridging the gap between law of large numbers and CLT.

Contribution

It introduces a Strassen's invariance principle for this model under subdiffusive variance conditions, expanding theoretical understanding.

Findings

Proves Strassen's invariance principle for random walk in random environment.

Uses martingale and fractional coboundary techniques.

Bridges the gap between law of large numbers and central limit theorem.

Abstract

In this paper, we consider random walk in random environment on $\mathbb{Z}^{d}\,(d\geq1)$ and prove the Strassen's strong invariance principle for this model, via martingale argument and the theory of fractional coboundaries of Derriennic and Lin \cite{DL}, under some conditions which require the variance of the quenched mean has a subdiffusive bound. The results partially fill the gaps between law of large numbers and central limit theorems.