An Almost Sure Invariance Principle for Renormalized Intersection Local Times
Richard F. Bass, University of Connecticut, USA
Jay Rosen, City University of New York, USA
Abstract
Let beta_k(n) be the number of self-intersections of order
k, appropriately renormalized, for a mean zero planar random walk
with 2+delta moments. On a suitable probability space we
can construct the random walk and a planar Brownian motion W_t such that for each k geq 2, |beta_k(n)- gamma_k(n)|=o(1), a.s.,
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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