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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Pages: 124-164

Publication Date: February 28, 2005

DOI: 10.1214/EJP.v10-236

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