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.
Categories: math.PR