Use the LIL (Corollary 12.2) to give an alternative proof for the fact that a onedimensional Brownian motion oscillates in every time interval \([0, \epsilon]\) infinitely often around its starting point.

Data From Corollary 12.2

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12.2 Corollary. Let (Bt)to be a BM. Then, almost surely, B(t) lim = 1, too2t log logt B(t) lim -1, t-00 2t log log t lim B(t) t-02t log log t| 1, B(t) lim -1. | to 2t log log t| Proof of Theorem 12.1. 1 (Upper bound). Fix e > 0, q> 1 and set sup B(s) (1+e)2q" log log an An = { sup From the maximal estimate (12.2), we see that P(An) 2P (B(q") > (1 + ) 2qn log log q B(q") = 2P (1+e)2 log log q" Since B(q)/q" ~B(1), we can use the upper bound from (12.1) and find P(An) < 1 (1+)log log q" e-(1+e) log log q"