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?
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$\begingroup$ This should be very closely related to the maximum process of Brownian motions, see e.g. Proposition 13.13 in Kallenberg's Foundations. $\endgroup$– Jochen WengenrothCommented Jan 8, 2014 at 7:38
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$\begingroup$ Yes, it is related to the reflection principle. In fact, we should better work with the probability for the process $W(t)$ to exit a sphere of radius $x$. Then we can modify the proof for one-dimensional Brownian motion so that the bound holds. Thank you. $\endgroup$– RaindogCommented Jan 8, 2014 at 10:44
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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).
answered Jan 13, 2014 at 17:06