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Suppose $P(X=1)=P(X=-1)=1/2$ and define

$$X_n=\begin{cases} X\ \text{with probability}\ 1-\frac{1}{n} \\ e^n\ \text{with probability}\ \frac{1}{n} \end{cases}$$

I then need to prove or disprove that $X_n$ converges to $X$ in probability, distribution and quadratic mean.

For the convergence in probability I noted that $P \left( |X_n-X|>\epsilon \right)=P\left( X_n>X+\epsilon \right)+P\left( X_n<X-\epsilon \right)=\frac{1}{n}+0 \to 0$

Assuming then that I got that right, $X_n$ also converges in distribution to $X$ as convergence in probability implies convergence in distribution.

I do not know how to handle the convergence in quadratic mean though. Could I please get some help there? If I can prove convergence in QM, the other implications follow, which makes me think that it won't be the case ;).

Thank you.

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