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

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

Richard F. Bass and Jay Rosen

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

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

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1127395876
Digital Object Identifier: doi:10.1214/009117905000000215
Mathematical Reviews number (MathSciNet): MR2165582
Zentralblatt MATH identifier: 1085.60018

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