From Khintchine’s law of the iterated logarithm (LIL), Corollary 12.2, we know that for a one-dimensional Brownian motion
Since Brownian paths are a. s. continuous, it is clear that the set of limit points of is the whole interval [-1,1].
This observation is the starting point for an extension of Khintchine’s LIL, due to Strassen , 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 ...