# convergence of random variables

• Dec 5th 2010, 04:59 PM
Beaky
convergence of random variables
Suppose $X_{n} \rightarrow X$ in probability and $X_{n}^{2} \le W, \forall n$ where $E(W)$ is finite. Show $X_{n} \rightarrow X$ in mean square.

So far I have:

$E((X_{n}-X)^2)=E(X_{n}^{2})-2E(X_{n}X)+E(X^{2})
\le E(W)-2E(X_{n}X)+E(X^{2})$

I'm not even sure if I'm on track or not. I can't think of how to use the convergence in probability to get anywhere.
• Dec 6th 2010, 09:35 AM
Moo
Hello,

Are you sure that it's $X_n^2\leq W$ ?
Have you studied Lebesgue's dominated convergence theorem ?

Yuck, this problem sucks !