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

For a symmetric random walk in $Z^2$ with $2+\delta$ 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 $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2\pi}}\log n +c_X)^{-j} \gamma_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $\gamma_{j,n}$ 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.