limit of random variables, expectation

Hallo!

Can anybody please help me?


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?

Thanks in advance!

Whoops, didn't read one of the conditions. I will think about this more.

OK. Thanks!

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 $\displaystyle \limsup \mathbb E X_h \le \mathbb E X \le \liminf \mathbb E X_h$).

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 :)

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?

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.

This question is making me very nervous.
But thank you so much for helping me,

Julia