0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.