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

An Almost Sure Invariance Principle for Renormalized Intersection Local Times