The Annals of Probability

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

Richard F. Bass and Jay Rosen

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Abstract

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.

Article information

Source
Ann. Probab., Volume 33, Number 5 (2005), 1856-1885.

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395876

Digital Object Identifier
doi:10.1214/009117905000000215

Mathematical Reviews number (MathSciNet)
MR2165582

Zentralblatt MATH identifier
1085.60018

Keywords
Range random walks invariance principle intersection local time Wiener sausage Brownian motion

Citation

Bass, Richard F.; Rosen, Jay. An almost sure invariance principle for the range of planar random walks. Ann. Probab. 33 (2005), no. 5, 1856--1885. doi:10.1214/009117905000000215. https://projecteuclid.org/euclid.aop/1127395876


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