TITLE: How to prove this Brownian motion convergence?
QUESTION [3 upvotes]: Let $W_t$ be a Brownian motion. How do I show the following?
$$
\alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.}
$$
Showing convergence of this in probability is easy enough, but I am not sure of how I could get the stronger a.s. result. By (super)martingale convergence I know that the limit is almost surely finite, but how do I show that it is equal to $0$?

REPLY [2 votes]: Exploiting the martingale property of the Brownian motion, we can show that for each positive $\varepsilon$, the inequality 
$$\mathbb P\left\{\sup_{2^n\leqslant t\leqslant  2^{n+1} }\left|W_t\right| \gt \varepsilon 2^{n\alpha} \right\}\leqslant 4 \varepsilon^{-2} 2^{n(1-2\alpha)},$$
holds. We then conclude by the Borel-Cantelli lemma.
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