I'm trying to solve the following:
Let $(a_n)_{n\in\mathbb{N}}$ be a bounded real value sequence and let $Y_n, n\in \mathbb{N}$ be i.i.d. random variables with $P(Y_n=1)=P(Y_n=-1)=1/2$. Let $$X_n=\sum\limits_{k=0}^nY_ka_k$$ and $\mathcal{F}_n=\sigma(X_0,\dots,X_n)$
- Show that $X_n$ converges in $L^2$ if and only iff $\sum\limits_{k=0}^\infty a_k^2<\infty$
- Compute $\langle X\rangle_n:=\sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1}-X_k^2|\mathcal{F}_k)$
My attempt for 1:
So it's easy to see that $(X_n)$ is a martingal $E(X_{n+1}\mid \mathcal{F}_n)=X_n+0=X_n$
Now we have to show $\lim\limits_{n\to\infty}\mathbb{E}(|X_n-X_{\infty}|^2)=0$ notice by Fatou $$\mathbb{E}(|X_n-\lim\limits_{k\to \infty}X_k|^2)\leq \liminf\limits_{k\to \infty}\mathbb{E}(|X_n-X_k|^2)=\sum\limits_{k=n+1}^\infty\mathbb{E}(|X_k-X_{k-1}|^2)$$
And here I'm getting stucked. If I compute $$\mathbb{E}(|X_k-X_{k-1}|^2)\leq \mathbb{E}(|X_k|^2+2|X_kX_{k-1}|+|X_{k-1}|^2)=1+2\mathbb{E}(|X_kX_{k-1}|)+1$$
Then I'm gettings obviously a sum that goes to infinity on the RHS.
My attempt for 2:
$\sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1}-X_k^2|\mathcal{F}_k)=\sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1}|\mathcal{F}_k)-X_k^2=\sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1})-X_k^2$ $$=\sum\limits_{k=0}^{n-1}\mathbb{E}\left(\left(\sum\limits_{k=0}^{k+1}Y_ka_k\right)^2\right)-X_k^2$$
And here I get again stucked.
$var(X_n)=\sum_{i=1}^na_i^2var(X_i)=\sum_{i=1}^na_i^2$ so if $\sum a_i^2<\infty$ then $\{X_n\}$ is $L^2$-bounded. Doob's $L^p$ martingale convergence theorem then gives an $L^2$ (and a.s.) limit. (Is this result available to you?) If $\sum a_i^2$ diverges then there can be no $L^2$ limit, $$E|X_n-X_\infty|^2=EX_n^2+EX_\infty^2-2E(X_nX_\infty)>EX_n^2-2||X_n||_2||X_\infty||_2+EX_\infty^2\to\infty $$ as $n\to\infty$ (and $EX_n^2=var(X_n)\to\infty$).
For the second part (the "predictable compensator") I think you were on the right track, $$ \sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1}-X_k^2|\mathcal{F}_k)=\sum\limits_{k=0}^{n-1}\mathbb{E}(X^2_{k+1}|\mathcal{F}_k)-X_k^2\\ =\sum\limits_{k=0}^{n-1}\mathbb{E}((X_k+Y_{k+1})^2\mid\mathcal{F}_k)-X_k^2= \sum\limits_{k=0}^{n-1}2X_k\mathbb{E}(Y_{k+1}\mid\mathcal{F}_k)+\mathbb{E}(Y_{k+1}^2\mid\mathcal{F}_k)\\ =n\cdot var(Y_1)=n. $$