Does convergence in mean imply convergence in mean square?

Question

I know that convergence in probability does not imply convergence in mean. However, does convergence in mean imply convergence in mean square? I can't think of any counter-examples of this so I don't know if it is true or not.

Answer 1

$X_n\overset{L^2}{{\to}}C$ ($X_n$ converges to C in mean square) if and only if

$$ \begin{cases} lim_nE(X_n)=C, \\ lim_nV(X_n)=0, \\ \end{cases}$$

Now consider the following sequence of rv's

$$ X_n=\begin{cases} \frac{n-1}{n}, & \text{if $X_n=0$ } \\ \frac{1}{n}, & \text{if $X_n=n$ } \end{cases}$$

Note that

$E(X_n)=1$ but

$lim_n V(X_n)=n-1=\infty$

Thus $X_n$ converges in $L^1$ but not in $L^2$