Ayanna Trujillo

2022-06-24

Problem:
Let $\left\{{X}_{i}{\right\}}_{i=1}^{\mathrm{\infty }}$ be a sequence of random variables on a probability space $\left(\mathrm{\Omega },\mathcal{F},P\right)$ such that a.e. Show that if $\underset{i}{sup}\text{E}\left({X}_{i}^{2}\right)<\mathrm{\infty }$, then $\text{E}\left({X}^{2}\right)<\mathrm{\infty }$.
My Attempt:
I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if $\left\{{X}_{i}{\right\}}_{i=1}^{\mathrm{\infty }}$ is a sequence of non-negative random variables, then $\text{E}\left(\underset{i}{lim inf}{X}_{i}\right)\le \underset{i}{lim inf}\text{E}\left({X}_{i}\right)$.
If we let ${Y}_{i}={X}_{i}^{2}$ then I have a sequence of non-negative random variables to work with. One concern of mine is this: can I assume that $\underset{i\to \mathrm{\infty }}{lim}{X}_{i}^{2}={X}^{2}$? I feel like that's necessary for for what I've written below to work.
If I can make that assumption then we have

Yes, your first step is justified. By the continuous mapping theorem, we have that if ${X}_{n}\to X$ almost surely, then $g\left({X}_{n}\right)\to g\left(X\right)$ almost surely for any continuous function. If we set $g\left(x\right)={x}^{2}$, then
${X}^{2}=\underset{n\to \mathrm{\infty }}{lim}{X}_{n}^{2}=\underset{n\to \mathrm{\infty }}{lim inf}{X}_{n}^{2}\phantom{\rule{1em}{0ex}}\text{a.s}$
where the last equality follows simply because the limit exists. The rest follows by taking expectations and applying Fatou's lemma.

Do you have a similar question?