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