Show that the proof of Khinchine's LIL, Theorem 12.1, can be modified to give \[\varlimsup_{t ightarrow \infty} \frac{\sup _{s \leqslant t}|B(s)|}{\sqrt{2 t \log \log t}} \leqslant 1\]

Use in Step \(1^{0}\) of the proof \(\mathbb{P}\left(\sup _{s \leqslant t}|B(s)| \geqslant xight) \leqslant 4 \mathbb{P}(|B(t)| \geqslant x)\).

Data From Theorem 12.1

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12.1 Theorem (LIL. Khintchine 1933). Let (B(t))to be a BM. Then P lim B(t) 2t log log t 1)=1. (12.3) Since (-B(t))20 and (tB(t))>o are again Brownian motions, we can immediately get versions of the Khintchine LIL for both the limit inferior and t 0. E