Let $(X_n\mid n=1,2,\ldots)$ be a sequence independent of random $2$-dimensional vectors, where, for each $n$, $X_n$ is uniformly distributed on the square with vertices $(\pm n,\pm n)$. Calculate the probability that $\left|X_n\right|\to\infty$ as $n\to\infty$.
Can somebody help me to solve this problem. I just don't know how to solve it or what kind of definition I should use.
I've trying to use the Borel-Cantanelli lemma but I'm kind of confused because the problem says that we are talking about a 2-dimensional vectors.
That means that we aro going to have $(X_1,X_2)$ a.v's independent or a sequence of different 2-dimensional vector. The Borel-Cantanelli lemma just talks about a sequence not a sequence of sequences. That is where I'm confused.