  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.
