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

2004/04/04 by Richard F. Bass, Bass, Richard F., Jay Rosen +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60G50 #60J65 #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60G50 #msc:60J65

paper · pdf · doi:10.48550/arxiv.math/0404070

arxiv created 2004/04/04 · openalex publication_date 2004/04/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 [ (1)/(n) |R(n)| +∑_j=1^k (-1)^j (\textstyle\frac12πlog n +c_X)^-j γ_j,n]→ 0, a.s. where W_t is a Brownian motion, W⁽ⁿ⁾_t=W_nt/√ 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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