Let $S_n$ be a random walk in $\bz^d$ and let $R_n$ be the range of $S_n$. We prove an almost sure invariance principle for $R_n$ when $d=3$ and a law of the iterated logarithm for $R_n$ when $d=2$.
"Law of the iterated logarithm for the range of random walks in two and three dimensions." Ann. Probab. 30 (3) 1369 - 1396, July 2002. https://doi.org/10.1214/aop/1029867131