Converging sucession of uniform variables.

Question

I have this question

Let $\{X_n\}_n$ a sucession of random variables such that $X_n\thicksim U([-n,n])$ with $n\geq 1$. In what sense the sucession $X_n$ converges to a random variable? (explain your solution)

I don't understand how it can converge to any random variable. Can anyone give me a hint or say me what to read to solve this correctly? I think that I need to use the moment-generating function, but when $n\mapsto \infty$ I get problems.

Answer 1

Observe that if $U$ is uniformly distributed on $[-1,1]$, then $X_n$ has the same distribution as $n\cdot U$. Therefore, the sequence $\left(X_n\right)_{n\geqslant 1}$ do not converge in distribution to any random variable.

What we can say is that $\left(1/X_n\right)_{n\geqslant 1}$ converges in probability to $0$.