Given a standard Brownian motion B, we have known that almost surely, $$\limsup_{n\to\infty}\frac{B(n)}{\sqrt{n}}=+\infty.$$

For any positive real number a and integer n, let $$E_n=\left\{\frac{B(n)}{\sqrt{n}}>a\right\}.$$ By CLT, we know that the measure of $E_n$ converges.

My question: Is there any good way to estimate the following term $$\mathbb{P}\left\{\bigcup_{k=n}^m E_k\right\}.$$

For simplify, if we take m=2n and fix a=1( or any number you want), I hope to know how to control the above measure in terms of n.

Thanks for any comments and suggestions.