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From Levy's continuity theorem it is known that pointwise convergence of characteristic functions is equivalent to the convergence in law of corresponding random variable. However, I have to show that if $X_n \sim\mathrm{Unif} [-n,n]$, then the characteristic functions of the sequence converge pointwise, but $(X_n)_{n \in \mathbb{N}}$ does not converge in law.

I have calculated the characteristic function which is $\frac{\sin(tn)}{tn}$ and which converges to 0. Still I do not know yet how to show, that the sequence of r.v. does not converge in law and more important, why this example does not contradict the stated theorem ?

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    The sequence of characteristic functions should converge pointwise to a characteristic function,not just converge somewhere. Then the Lavy Cont. Thm. says that the sequence $X_n$ will converge in law to a r.v. $X$ which has characteristic function as the limiting characteristic function. In your example, $0$ is not a characteristic function, so Levy does not hold. – 2017-01-21
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    Of course you're right ! Thanks for the hint. – 2017-01-21

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