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!