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 $\mathrm{\mid }\mathcal{R}(n)\mathrm{\mid }$, 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 $k\ge 1$ $$(\log n{)}^k[\frac{1}{n}\mathrm{\mid }\mathcal{R}(n)\mathrm{\mid }+\sum _{j=1}^k(-1{)}^j(\frac{1}{2\pi }\log n+c_X{)}^{-j}{\gamma }_{j,n}]\to 0,a\mathrm{.}s\mathrm{.}$$ where $W_t$ is a Brownian motion, $W_t^{(n)}=W_{nt}\mathrm{/}\sqrt{n}$, ${\gamma }_{j,n}$ 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.