Properties about a certain martingale I asked this question here. Unfortunately there was not a satisfying answer. So I hope here is someone who could help me.
I'm solving some exercises and I have a question about this one:
Let $(X_i)$ be a sequence of random variables in $ L^2 $ and a filtration $ (\mathcal{F}_i)$ such that $X_i$ is $\mathcal{F}_i$ measurable. Define
$$ M_n := \sum_{i=1}^n \left(X_i-E(X_i|\mathcal{F}_{i-1})\right) $$
I should show the following:

*

*$M_n $ is a martingale.

*$M_n $ is square integrable.

*$M_n $ converges a.s. to $ M^*$ if $ M_\infty := \sum_{i=1}^\infty E\left((X_i-E(X_i|\mathcal{F}_{i-1}))^2|\mathcal{F}_{i-1}\right)<\infty$ .

*If $\sum_{i=1}^\infty E(X_i^2) <\infty \Rightarrow 3)$
I was able to show 1 and with Davide Giraudo's comment 2. is clear too. But I got stuck at 3. and 4. So I'm very thankful for any help!
hulik

For 3, compute $E[M_n^2]$. Having done this conclude that $E[M_n^2]\le M_\infty$ for all $n$. This means that $\{M_n\}$ is an $L^2$-bounded martingale, to which the martingale convergence theorem may be applied.
For 4, the $i$th term in the sum defining $M_\infty$ is equal to
$E[(X_i-E[X_i|\mathcal{F}_{i-1}])^2]$, which in turn is equal to
$E[X_i^2] -E[(E[X_i|\mathcal{F}_{i-1}])^2]\le E[X_i^2]$.
