An almost sure invariance principle for renormalized intersection local times

Details An almost sure invariance principle for renormalized intersection local times An almost sure invariance principle for renormalized intersection local times

2004-07-08

arxiv.org/abs/math/0407149v1

Mathematics

Probability

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Let \beta_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+\delta moments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for some a>0 where \gamma_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.

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