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I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:

Let $B_t$ a brownian motion, then for every $a \in \mathbb{R} \ $,

$$\mathbb{P}(\lim_{t \to \infty} \sup_{s\in [0,t]} B_s > a) = 1$$

I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.

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  • $\begingroup$ Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := \sup_{s \leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion. $\endgroup$ – saz Dec 7 '18 at 7:51

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