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 Z2 with 2+δ moments, we represent |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 Wt is a Brownian motion, $W^{(n)}_{t}=W_{nt}/\sqrt{n}$, γj,n is the renormalized intersection local time at time 1 for W(n) and cX 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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