TITLE: Speed of covering the circle with a random interval
QUESTION [0 upvotes]: 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?

REPLY [1 votes]: Hint: it suffices to do the cases $x=0$ and $x=1$.
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