# Urgent homework on expectation

• Sep 23rd 2008, 05:47 AM
UMD
Urgent homework on expectation
Can anyone help me in proving the following question

Let {Xn} be a sequence of random variables satisfying Xn <= Y a.s for some Y with E(|Y|) < infinity. then show that

E( Lim n approaches infinity Sup Xn) > = lim n approaches infinity Sup E (Xn)

I think that the all you have to do is to apply Fatou's lemma to $Y-X_n$ (which is non-negative), and to use the fact that $\limsup_n (a-u_n)=a-\liminf_n u_n$.
Notice as well that $E[X_n]$ makes sense since the positive part of $X$ is integrable ( $X_+\leq |Y|$), so that you can write $E[Y-X_n]=E[Y]-E[X_n]$ (where infinite values are possible).