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September 2005 An almost sure invariance principle for the range of planar random walks
Richard F. Bass, Jay Rosen
Ann. Probab. 33(5): 1856-1885 (September 2005). DOI: 10.1214/009117905000000215


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.


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Richard F. Bass. Jay Rosen. "An almost sure invariance principle for the range of planar random walks." Ann. Probab. 33 (5) 1856 - 1885, September 2005.


Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1085.60018
MathSciNet: MR2165582
Digital Object Identifier: 10.1214/009117905000000215

Keywords: Brownian motion , Intersection local time , invariance principle , Random walks , ‎range‎ , Wiener sausage

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • September 2005
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