Law of Large numbers using Brownian limit

Question

Given a standard Brownian motion $\{B_t;0 \leq t < \infty \}$, we know that $\lim_{t \to \infty}\frac{B_t}{t} = 0$ a.s. I am interested to know if we can prove Strong Law of Large Numbers for any finite second moment IID variables using the above result. One of the ways I was looking at it is to use Skorokhod embedding. If $X_1,X_2,X_3,...$ are IID random variables with zero mean and unit variance we know that we can find stopping times $T_1, T_2, ...$ such that $B_{T_1}, B_{T_2} - B_{T_1},... $ are IID with same distribution as $X_i$. If we can prove that $T_n \to \infty$ then using this we can establish SLLN. Can some one help me with proving $T_n \to \infty$. Thanks

Answer 1

$(T_{k+1}-T_k)_{k=1,2,\cdots}$ is an IID sequence. Let $X=T_2-T_1$. Since $\mathbb{P}[X=0]<1$, there is a positive number $\epsilon$ such that $\mathbb{P}[X>\epsilon]>0$. So, by independence and the second Borel-Cantelli lemma, $\mathbb{P}[T_{k+1}-T_k >\epsilon \text{ infinitely often}] =1$. If $T_{k+1}-T_k >\epsilon$ infinitely often, then $T_n \to \infty$ as $n\to\infty$. Thus $T_n\to\infty$, almost surely.