Moderate Deviations for the Range of Planar Random Walks

Given a symmetric random walk in ${mathbb Z}^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. The authors study moderate deviations for $R_n -{mathbb E}R_n$ and ${mathbb E}R_n -R_n$. They also derive the corresponding laws of the iterated logarithm.