$|W_{t_n}| \to \infty$ a.s. as $n \to \infty$ for Brownian $(W_t)_{t \in \mathbb{R}_+}$ with $\sum_{n=1}^\infty t_n^{-0.5} < \infty$?

Question

Let $(W_t)_{t \in \mathbb{R}_+}$ be a Brownian motion. Let $(t_n)_{n \in \mathbb{N}} \subset \mathbb{R}_+$ be a sequence of time points, such that $\sum_{n=1}^\infty t_n^{-0.5} < \infty$ (so, for example, the sequences $t_n = n$ or $t_n = n^2 $ are not increasing sufficiently fast, but the sequence $t_n = n^3$ is increasing fast enough). Prove that \begin{align*} |W_{t_n}| \to + \infty \quad \text{a.s. as } n \to +\infty \end{align*}

I stumbled across this problem in a relatively unknown stochastic process textbook and I have no idea how to solve it. Thank you in advance for your help.

Answer 1

One can use the Borel-Cantelli lemma, reducing the problem to show that for each positive $\varepsilon$, $\sum_{n=1}^\infty \mathbb P\left(1/\lvert W_{t_n} \rvert >\varepsilon\right)$ converges. To do so, one can express $\mathbb P\left(1/\lvert W_{t_n} \rvert >\varepsilon\right)$ as the probability that a standard normal random variable lies between the bounds $\pm t_n^{-1/2}/\varepsilon$. Then using the fact that the density of such a random variable is bounded, one gets the result from the convergence $\sum_{n\geqslant 1}t_n^{-1/2}$.