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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?

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Hint: The reflection principle gives that $M_t$ has the same distribution as $|B_t|$. –  Chris Janjigian May 2 '12 at 23:46
    
Sorry I used $B_t$ as my notation for a Brownian motion there. –  Chris Janjigian May 2 '12 at 23:52
    
It also might help you at some point when estimating the tail of the normal distribution that there is a really stupid bound: $e^{-x^2}\leq xe^{-x^2}$ for $x$ large. –  Alex R. May 3 '12 at 0:27
    
So you are saying to use Hoeffdings inequality? will it help? –  mary May 3 '12 at 1:59
    
Perhaps I was being too cryptic. Chris suggested that the distribution of $M_t$ is the same as $|B_t|$. You can easily calculate the probability that $|B_t|\geq at$ for $a$ a positive constant, as a Gaussian integral but you'll need to do some estimates for how this "tail" probability behaves for large $t$. I believe my prior hint will help. –  Alex R. May 3 '12 at 2:53
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1 Answer 1

Since $B_t$ is symmetric, the result recalled in the post reads $\mathrm P(M_t\gt at)=\mathrm P(|B_t|\gt at)$ $(\ast)$.

Since $B_t$ equals $\sqrt{t}B_1$ in distribution, $(\ast)$ is $\mathrm P(|B_1|\gt a\sqrt{t})$. Since $|B_1|$ is almost surely finite and $a\sqrt{t}\to+\infty$ for every $a\gt0$, the last probability goes to zero, hence $\mathrm P(M_t/t\gt a)\to0$ for every $a\gt0$. That is, $M_t/t\to0$ in probability.

One can strengthen this convergence, noting that the series $\sum\limits_n\mathrm P(|B_1|\gt a\sqrt{n})$ converges since $\mathrm P(|B_1|\gt a\sqrt{n})$ is $\mathrm e^{-\frac12a^2n+o(n)}$. By the first Borel-Cantelli lemma, $M_n/n\to0$ almost surely when the integer $n$ goes to infinity. For every $t\geqslant1$ there exists an integer $n\geqslant1$ such that $n\leqslant t\lt n+1$, and $(M_t)_t$ is nondecreasing hence $M_t/t\leqslant M_{n+1}/n\leqslant 2M_{n+1}/(n+1)$. This proves that $M_t/t\to0$ almost surely when the real number $t$ goes to infinity.

The same identity $(\ast)$ shows the convergence in $L^p$ for every finite $p\geqslant1$.

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