convergence of modulus BM under a subsequence

Question

Is it true that if $B$ is a BM, then there exists a subsequence $t_n$ such that $|B_{t_n}| \to \infty$ almost surely? Does this follow from the law of the iterated logarithm directly?

Answer 1

It does not follow immediately from LIL. LIL only gives you a sequence depending on $\omega$.

Consider $P(\frac 1 {|B_{t_n}|} >\epsilon) =P(|B_{t_n}| <\frac 1 {\epsilon})$. Since $B_{t_n} = \sqrt t_n X$ where $X \sim N(0,1)$ we get $\sum_n P(\frac 1 {|B_{t_n}|} > \epsilon)=\sum_n P(|X| <\frac 1 {\epsilon \sqrt {t_n}}) \leq \sum_n \frac 2 {\sqrt {2\pi} \epsilon \sqrt {t_n}}$ Using the fact that $e^{-x^{2}/2} \leq 1$. This last sum is finite if $t_n >2^{2n}$. By Borel - Cantelli Lemma we see that $|B_{t_n}| \to \infty $ almost surely.