Ayanna Trujillo

2022-06-24

Problem:

Let $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ be a sequence of random variables on a probability space $(\mathrm{\Omega},\mathcal{F},P)$ such that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}=X\text{a.e.}$ a.e. Show that if $\underset{i}{sup}\text{E}({X}_{i}^{2})<\mathrm{\infty}$, then $\text{E}({X}^{2})<\mathrm{\infty}$.

My Attempt:

I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ is a sequence of non-negative random variables, then $\text{E}(\underset{i}{lim\u2006inf}{X}_{i})\le \underset{i}{lim\u2006inf}\text{E}({X}_{i})$.

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

$\begin{array}{rl}\text{E}({X}^{2})& =\text{E}(\underset{i}{lim\u2006inf}{X}_{i}^{2})\mathbf{\text{(Is this justified?)}}\\ & \le \underset{i}{lim\u2006inf}\text{E}({X}_{i}^{2})\text{(application of Fatou's Lemma)}\\ & \le \underset{i}{sup}\text{E}({X}_{i}^{2})\text{(property of real numbers)}\\ & \mathrm{\infty}\text{(by assumption)}.\end{array}$

Let $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ be a sequence of random variables on a probability space $(\mathrm{\Omega},\mathcal{F},P)$ such that $\underset{i\to \mathrm{\infty}}{lim}{X}_{i}=X\text{a.e.}$ a.e. Show that if $\underset{i}{sup}\text{E}({X}_{i}^{2})<\mathrm{\infty}$, then $\text{E}({X}^{2})<\mathrm{\infty}$.

My Attempt:

I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if $\{{X}_{i}{\}}_{i=1}^{\mathrm{\infty}}$ is a sequence of non-negative random variables, then $\text{E}(\underset{i}{lim\u2006inf}{X}_{i})\le \underset{i}{lim\u2006inf}\text{E}({X}_{i})$.

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

$\begin{array}{rl}\text{E}({X}^{2})& =\text{E}(\underset{i}{lim\u2006inf}{X}_{i}^{2})\mathbf{\text{(Is this justified?)}}\\ & \le \underset{i}{lim\u2006inf}\text{E}({X}_{i}^{2})\text{(application of Fatou's Lemma)}\\ & \le \underset{i}{sup}\text{E}({X}_{i}^{2})\text{(property of real numbers)}\\ & \mathrm{\infty}\text{(by assumption)}.\end{array}$

hildiadau0o

Beginner2022-06-25Added 21 answers

Yes, your first step is justified. By the continuous mapping theorem, we have that if ${X}_{n}\to X$ almost surely, then $g({X}_{n})\to g(X)$ almost surely for any continuous function. If we set $g(x)={x}^{2}$, then

${X}^{2}=\underset{n\to \mathrm{\infty}}{lim}{X}_{n}^{2}=\underset{n\to \mathrm{\infty}}{lim\u2006inf}{X}_{n}^{2}\phantom{\rule{1em}{0ex}}\text{a.s}$

where the last equality follows simply because the limit exists. The rest follows by taking expectations and applying Fatou's lemma.

${X}^{2}=\underset{n\to \mathrm{\infty}}{lim}{X}_{n}^{2}=\underset{n\to \mathrm{\infty}}{lim\u2006inf}{X}_{n}^{2}\phantom{\rule{1em}{0ex}}\text{a.s}$

where the last equality follows simply because the limit exists. The rest follows by taking expectations and applying Fatou's lemma.

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