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

• 成果类型: Article
• 署名作者: Bass, RF; Rosen, J
• 署名单位: University of Connecticut; City University of New York (CUNY) System; College of Staten Island (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000215
• 发表日期: 2005
• 页码: 1856-1885
• 关键词: intersection local-times; recurrent random-walks; iterated logarithm; random-variables; brownian-motion; Levy processes; LAWS

摘要

For a symmetric random walk in Z(2) with 2 + delta moments, we represent vertical bar R(n)vertical bar, 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 [Graphics] where Wt is a Brownian motion, W(t)((n)) = Wnt/root n, gamma 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.