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

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

2004-04-04

arxiv.org/abs/math/0404070v1

Mathematics

Probability

Scientific paper

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.

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