TITLE: Deviation bound for the maximum of the norm of Wiener process
QUESTION [6 upvotes]: Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof:
$$
{\mathbf P} \left\{ \max_{t\in[0,2]} \|W(t)\|> x\right\}\leqslant 2 {\mathbf P} \bigl\{\chi_n^2>x^2/2\bigr\}.
$$
Why is this bound true? What other deviation inequalities for the Bessel processes are known?

REPLY [3 votes]: A general bound that's sometimes useful is given by Fernique's theorem.  It's a general fact about Gaussian measures on Banach spaces, but in this case it gives the following: there are constants $C, \epsilon > 0$ (depending on the dimension $n$) such that
$${\mathbf P} \left\{ \max_{t\in[0,2]} \|W(t)\|> x\right\}\leqslant C e^{-\epsilon x^2}.$$
You can find a proof in these lecture notes of mine (see Section 4.2).