Let ($X_n$)$_{n \in \mathbb N}$ be a sequence of random variables defined on a probability space ($\Omega,F,P$). Suppose that $X_n \geq 0$ and $E(X_n ^4) < \frac{1}{n^\frac{3}{2}}$
, for all $n \in \mathbb N$. Show that $X_n$ converges to $0$ almost surely as $n \rightarrow \infty $
I know I need to show that $P(X_n \rightarrow 0) = 1$ as $n \rightarrow \infty$
I have no idea how to show this though especially because I don't know what $\Omega$ is.
Best Answer
A few things that you should note:
1). In most circumstances you will never know, nor do you need to know what $\Omega$ is and other things about $\Omega$ like what is it's topology. So, if you are thinking of applying a set-theoratic or topological argument on $\Omega$, then it won't work in Measure Theoratic Problems like this.
2). A sufficient condition for a sequence of random variables $X_{n}$'s to converge almost surely to $0$, is that $\sum_{n}P(|X_{n}|>\epsilon)<\infty$ for each $\epsilon>0$. The proof of this follows from the Borel-Cantelli Lemma.
So what can you say about $P(|X_{n}|>\epsilon)=P(|X_{n}|^{4}>\epsilon^{4})$ by applying Markov's Inequality?
3). If you don't want to use the above, then you can follow the hint by Nejimban in the comments.
You first argue that $E(\sum_{n=1}^{\infty}X_{n})=\sum_{n=1}^{\infty}E(X_{n})$ by an application of Tonneli's Theorem.
Now since the RHS is finite, you have shown that $\sum_{n=1}^{\infty}X_{n}$ is almost surely finite (else it wouldn't have had finite expectation).
This means that for all $\omega\in\Omega$ except for a set $A$ of $0$ probability, you have that $\sum_{n=1}^{\infty}X_{n}(\omega)<\infty$.
Now recall your real-analysis notes. Does this imply that $X_{n}(\omega)\to 0$? If not, then can the sum be finite.
Now you should check the definition for convergence almost surely and check if the work you did above implies that $X_{n}$ converges almost surely to $0$ or not.
Note: I have not used any property of $\Omega$ except for the fact that it is simply a set and has elements in it called $\omega$ for this problem.