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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:

  1. $M_n $ is a martingale.
  2. $M_n $ is square integrable.
  3. $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$ .
  4. 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

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    Since $X_i$ and $E(X_i\mid \mathcal F_{i-1})$ are $L^2$, $M_n$ is a $L^2$ martingale. In fact, $M_{\infty}$ is the sum for $\left[M,M\right]_n$ where $\left[M,M\right]_n$ is the bracket of $M$ (such that $\{M_n^2-\left[M,M\right]_n\}$ is a $(\mathcal F_n)$-martingale).2011-12-10
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    Hm...ok that was not to hard. But what's about 3? I have seriously no idea how to prove this. Some help would be appreciated.2011-12-11
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    What you mean by bracket? Do you mean quadratic variation? Why should it have this form? Looking at the formal definition on Wikipedia it looks different. Sorry I'm not very familiar with this. If it is the quadratic variation, is there a theorem, that if it's finite then it converge a.s.?2011-12-14

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