Strassen's invariance principle for random walk in random environment

In this paper, we consider random walk in random environment on ${\mathbb{Z}}^d(d\ge 1)$ 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.