# conditional expectation projection property

I'm currently try to understand the properties of the conditional expectation. Consider a probability space $$(\Omega, \mathcal{A}, \mathbb{P})$$ and an iid sequence of random variables $$(\varepsilon_j)_{j \in \mathbb{Z}}$$ on this probability space and let $$\xi_j = (\dots,\varepsilon_{j-1},\varepsilon_j)$$. Further let $$g$$ be a measurable function such that $$X_j = g(\xi_j)$$ describes a well defined sequence of random variables with mean zero. Next let $$F = (\mathcal{F}_k)_{k \in \mathbb{Z}} = (\sigma(\xi_k))_{k \in \mathbb{Z}}$$ be the filtration of sigma-algebras generated by $$\xi_k$$.

I want to prove, that the following representation is valid:

$$X_k = \sum_{l \in \mathbb{Z}} \mathbb{E}(X_k \mid F_l) - \mathbb{E}(X_k \mid F_{l-1})$$.

So far, I was able to obtain so far is that $$\mathbb{E}(X_k \mid F_l) - \mathbb{E}(X_k \mid F_{l-1}) = 0$$ for all $$l > k$$ and hence we can write

$$\sum_{l \in \mathbb{Z}} \mathbb{E}(X_k \mid F_l) - \mathbb{E}(X_k \mid F_{l-1}) \\= \sum_{l=-\infty}^{k} \mathbb{E}(X_k \mid F_l) - \mathbb{E}(X_k \mid F_{l-1}) \\= \lim_{n\to \infty} \sum_{l=-n}^{k} \mathbb{E}(X_k \mid F_l) - \mathbb{E}(X_k \mid F_{l-1}) \\ = \lim_{n\to \infty} (\mathbb{E}(X_k \mid F_k) - \mathbb{E}(X_k \mid F_{n-1}))$$.

We know that $$\mathbb{E}(X_k \mid F_k) = X_k$$. Hence it only remains to treat the remainder term. I was wondering, if I could use Levy's downwards theorem to obtain

$$\lim_{n\to \infty} \mathbb{E}(X_k \mid F_{n-1}) = \mathbb{E}(X_k \mid F_{-\infty})$$, where $$F_{-\infty}$$ is the intersection of all $$F_k$$. Then – at least I assume – we have $$F_{-\infty} = \{\emptyset, \Omega\}$$ the trivial sigma algebra und thus $$\mathbb{E}(X_k \mid F_{-\infty}) = \mathbb{E}(X_k) = 0$$ and thus the proof would be complete.

Would someone share his/her thoughts on this?

$$F_{-\infty}$$ need not be $$\{\emptyset, \Omega\}$$ but any set in this $$\sigma-$$ field has probability $$0$$ or $$1$$ and this is good enough for your conclusion. Apply Kolmogorobv's $$0-1$$ law to the sequence $$\epsilon_{-1},\epsilon_{-2},\epsilon_{-3},\cdots$$ to see that any set in $$F_{-\infty}$$ has probability $$0$$ or $$1$$.
• But I still need Levy's downwards theorem for the convergence of the sequence $\lim_{n\to \infty} \mathbb{E}(X_k \mid F_{n-1}) = \mathbb{E}(X_k \mid F_{-\infty})$, don't I? Apr 23, 2022 at 11:45