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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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13 Strassen’s functional law of the iterated logarithm

From Khintchine’s law of the iterated logarithm (LIL), Corollary 12.2, we know that for a one-dimensional Brownian motion

e9783110307290_i2856.jpg

Since Brownian paths are a. s. continuous, it is clear that the set of limit points of e9783110307290_i2857.jpg is the whole interval [-1,1].

This observation is the starting point for an extension of Khintchine’s LIL, due to Strassen [211], which characterizes the set of limit points as s → ∞ of the family

denotes the set of all continuous functions ω : [0,1] → with ω(0) = 0.) If is a limit ...

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