If $X$ is a standard 1d brownian motion and $M_t$ $= \mbox{max}\{X_S: 0 \le s \le t\} $, what can we say about $M_t/t$ as $t \rightarrow \infty$?
Mainly, what can we say about the behavior of this martingale?
My attempt: P(Msubt > a) = 2P(B(t) >a), but what integral does this fit in with?