An almost sure invariance principle for the range of planar random walks

Abstract

For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal {R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k≥1$ $$(\log n)^{k}\Biggl[\frac{1}{n}|\mathcal{R}(n)|+\sum_{j=1}^{k}(-1)^{j}\biggl(\frac{1}{2\pi}\log n+c_{X}\biggr)^{-j}\gamma_{j,n}\Biggr]\to 0\qquad\mbox{a.s.,}$$ where $W_t$ is a Brownian motion, $W^{(n)}_{t}=W_{nt}/\sqrt{n}, γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$ and $c_X$ is a constant depending on the distribution of the random walk.