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
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 $Z_1$.