Can anybody help me to show the first inequality in this proof? It is about standard Brownian Motion. So B(t) is a Brownian Motion and I know the rescaling and shifting properties of Brownian Motions but I don't know how to handle the supremum and how to get the $2^n m^2$ out of the $\mathbb{P}$?

The second inequality is clear.

Would be great if somebody could give me a hint!

Thanks for your help.


This is just an application of the union bound: $$P\left(\sup \limits_{i = 1}^n X_i > c\right) = P\left(\bigcup \limits_{i = 1}^n \{X_i > c\}\right) \le \sum \limits_{i = 1}^n P(X_i > c) \le n \sup \limits_{i = 1}^n P(X_i > c)$$

  • $\begingroup$ Thank you! This was very helpful! $\endgroup$ – nabla Dec 7 '16 at 20:18

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