[Math] Law of large numbers for Brownian Motion

Question

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold?

$$\lim_{n \to \infty}\frac{B_{T_n}}{T_n} = 0$$ a.s.

Answer 1

This is an almost sure property hence the result you are asking to check is equivalent to the following.

Let $f:\mathbb R_+\to\mathbb R$ denote a function such that $\lim\limits_{t\to+\infty}f(t)=0$ and $(t_n)$ a sequence of nonnegative real numbers such that $\lim\limits_{n\to\infty}t_n=+\infty$. Then $\lim\limits_{n\to\infty}f(t_n)=0$.

Surely you can prove this. Then fix some $\omega$ in $\Omega$, consider the function $f$ defined by $f(t)=B_t(\omega)/t$ and the real numbers $t_n=T_n(\omega)$, and apply the deterministic result.