An almost sure invariance principle for renormalized intersection local times
Abstract
Let βk(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk Xn in Z2 with 2+δ moments. On a suitable probability space we can construct Xn and a planar Brownian motion Wt such that for each k≥ 2, |βk(n)-γk(n)|=O(n-a), a.s. for some a>0 where γk(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion Wnt/ n.
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