Speed of covering the circle with a random interval

Question

Let $x\in [0,1)$, $L_x = x + \mathbb Z_+$ and $(t_k)_k$ be a sequence of i.i.d exponentially distributed random variables with parameter $\lambda>0$.

Then I think that

$\lim_{k\rightarrow\infty}\left[\frac{1}{k} \#\left\{ \ [0,\sum_{j=1}^k t_k]\cap L_x \right\}\right]$

exists a.s., is independent of $x$ and equals $\frac{1}{\lambda}$. This should follow from the law of large numbers saying that $\frac{1}{k}\sum_{j=1}^k t_j \rightarrow 1/\lambda$ a.s.

How can I make this precise?

Answer 1

Hint: it suffices to do the cases $x=0$ and $x=1$.