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Suppose I have a random variable $Z$ defined on a probability space built on a $\sigma$-algebra $\mathcal{F}$ (with $\mathbb{E}[Z^2] < \infty$). Moreover, suppose that I have two filtrations $\mathcal{A}$, $\mathcal{B} \subseteq \mathcal{F}$. My question is: If we define the random variables $$ X = \mathbb{E}[Z | \mathcal{A}] \text{ and } Y = \mathbb{E}[Z | \mathcal{B}], $$ when can we guarantee that $X$ and $Y$ are positively correlated?

In all the examples where I've constructed $\mathcal{F}$ by hand, it is easy to verify that $X$ and $Y$ are positively correlated. But I'm hoping that there's a generic argument out there showing that this property always holds (provided that $\mathcal{F}$ is nice in some way).

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Of course not. Let $X,Y$ be negatively correlated centered standard Gaussian (say with correlation -1/2). Let $Z=X+Y$ and let ${\cal A}=\sigma(X)$, ${\cal B}=\sigma(Y)$. Then $E(Z|{\cal A})=X+E(Y|{\cal A})=X-X/2=X/2$, and $E(Z|{\cal B})=Y/2$.

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