# limit of random variables, expectation

• Mar 25th 2011, 06:27 AM
Juju
limit of random variables, expectation
Hallo!

Is the following equality true?
$\displaystyle \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]$

$\displaystyle X$ is a random variable $\displaystyle X>0$
$\displaystyle X_h$ is a sequence of random variables, $\displaystyle 1>h>0, X_h>0$

with
$\displaystyle \mathbb{E}[X_h] \leq \mathbb{E}[X]$
$\displaystyle p\lim\limits_{h \to 0}X_h=X,$ where $\displaystyle p$ denotes convergence in probability

Now, I think it follows with Fatou's Lemma for non-negative random variables, that
$\displaystyle \liminf\limits_{h \to 0}\mathbb{E}[X_h]\geq \mathbb{E}[\liminf\limits_{h \to 0}X_h]$.

Can I now say, that
$\displaystyle \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]$
is true?

• Mar 26th 2011, 07:33 AM
theodds
• Mar 26th 2011, 07:40 AM
Juju
OK. Thanks!
• Mar 26th 2011, 08:19 AM
theodds
The issue I have with your logic is the invocation of Fatou's Lemma. You should need almost sure convergence for that. Everything else goes through fine I think (if you have Fatou's Lemma then you can get $\displaystyle \limsup \mathbb E X_h \le \mathbb E X \le \liminf \mathbb E X_h$).
• Mar 26th 2011, 08:23 AM
theodds
Okay, Fatou's Lemma looks fine. Apparently all you need is convergence in probability for that. Sorry for any confusion my ignorance may have caused :)
• Mar 26th 2011, 08:28 AM
Juju
So, you think under the given assumptions
$\displaystyle \lim\limits_{h \to 0}EX_h=EX$ is correct?

But do you have any idea how to show it?
• Mar 26th 2011, 09:47 AM
theodds
Okay, this question is starting to make me quite nervous. The first inequality I posted should come from the fact that $\displaystyle \mathbb E X_h \le \mathbb E X$ and the second inequality is the version of Fatou's Lemma I was able to find that concerned convergence in probability - that is, $\displaystyle \mathbb E X \le \liminf \mathbb E X_h$ provided we have convergence in probability, i.e. you can get rid of the liminf.
• Mar 26th 2011, 09:52 AM
Juju
This question is making me very nervous.
But thank you so much for helping me,

Julia