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I've been facing the following problem:

Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $

Verify if the following random variable sequence converges:

$ \frac{1}{n} \sum \limits_{k=1}^n \textbf{1}_{\{X_n < Y_n\}} $

So what I have reached so far:

If we fix $ \omega \in \Omega $, by Stolz theorem:

$ \lim\limits_{n \rightarrow \infty} \frac{a_1 + \dots + a_n}{n} = \lim\limits_{n \rightarrow \infty} a_n $, thus:

$ \lim\limits_{n \rightarrow \infty} \frac{1}{n} \sum \limits_{k=1}^n \textbf{1}_{\{X_n < Y_n\}} = \lim\limits_{n \rightarrow \infty} \textbf{1}_{\{X_n < Y_n\}} $.

And there goes the difficulty: if I am to verify if this sequence converges almost sure, I need to know what it might converge to. What might be the limit of such sequence? And if there is none, how to prove it?

Thanks in advance

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    $\begingroup$ Ok, so I got this: Let's define a new variable: $Z_n = 1 \iff X_n < Y_n $ and $ 0 $ otherwise Then, by large numbers law ($Z_n$ is iid: $ \lim\limits_{n \rightarrow \infty} \frac{1}{n}\sum\limits_{k=1}^n \textbf{1}_{\{Z_k = 1\}} = \mathbb{E}Z_1 = \frac{1}{2} $ $\endgroup$ – Jytug Jun 3 '14 at 19:24
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Indeed, the key point is to apply the law of large numbers to the sequence $(Z_j)_{j\geqslant 1}:=(\chi_{\{X_j\lt Y_j\}})_{j\geqslant 1}$. Since $(X_k,Y_k)_k$ is independent, so is $(Z_j)_{j\geqslant 1}$. Since $X_j\lt Y_j\Leftrightarrow X_j/j\lt Y_j/j$ and the random variables $ X_j/j$ and $Y_j/j$ are independent and uniformly distributed on the unit interval, $Z_j$ has the same distribution as $Z_1$.

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