I'm struggling with the following problem. Let $X_n$ be a sequence of non-negative random variables which are finite almost surely and all with expectation 1. Assume they converge in distribution against some other random variable $X$. Show that $X$ has expectation smaller or equal 1.
I first thought that Fatou would help, but this is only helpful for the almost everywhere convergence, isn't it?
Thank you in advance,
Fischaaa
We have for any non-negative random variable $Y$ that $$\mathbb E\left[Y\right]=\int_0^{+\infty}\mathbb P\left(\left\{Y\gt t\right\}\right)\mathrm dt.$$ Define $f_n(t) :=\mathbb P\left(\left\{X_n\gt t\right\}\right)$ and $f(t) :=\mathbb P\left(\left\{X\gt t\right\}\right)$ on $[0,+\infty)$ (that we endowed with the Borel $\sigma$-algebra and Lebesgue measure $\lambda$). By the convergence in distribution of $\left(X_n\right)_{ n\geqslant 1}$ to $X$, we have $f_n(t)\to f(t)$ at each point of continuity of $f$. The set of discontinuity points of $f$ is at most countable, hence $f_n(t)\to f(t)$ for $\lambda$-almost every $t$. Now we can conclude from Fatou's lemma.