Strong Law of Large Numbers converge a.s.

Question

The Question is:

Let ($X_{n})_{n≥1}$ be a sequence of i.i.d. Bernoulli random variables, on the same probability space, with parameter $\frac{1}2$ (P($X_{n}$ = 0) = P($X_{n}$ = 1) = $\frac{1}2$),and let $t_{n}$ be the hitting time of level n by the partial sums, i.e. $t_{n}$ = inf{k | $\mathbf{\Sigma^{k}_{m=1}}X_{m} = n$}. Show that $n^{-1}t_{n}$ converges to 2 almost surely.

My instinct here is to use the Strong Law of Large Numbers and use the

$lim_{n\rightarrow\infty} \ \frac{S_{n}}n = E({X_{1}}) = p\ \ a.s. $

But I am lost here. If we use the above, wouldn't I get that it converges to $\frac{1}2$ almost surely? Please help!

Answer 1

Notice that inter-arrival times $(t_i - t_{i-1})_{i=1}^{\infty}$ form an i.i.d. sequence of geometric random variables of parameter $p = 1/2$. Thus by SLLN,

$$ \frac{t_n}{n} \to \Bbb{E}[t_1] = \frac{1}{p} = 2 \quad \text{a.s.} $$