Abstract
Although we have informally thought of Brownian motion as a continuous approximation to random walks in discrete time, the approximations of Chapters IV and V, which are developed more completely in Chapter IX, do not actually utilize this idea. For example, Theorem 9.54 and Corollary 9.55, which are used as partial justification for (4.40), (4.41), (4.49), and (4.50), involve probabilities of large deviations, which can be quite different for Brownian motion and for random walk in discrete time.
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