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Let $(\Omega, F, P)$ be our probability space. Denote $\Omega^{(n)}$ to be a partition of $\Omega$ with $n$ components/sets such that each component has equal probability measure. Denote the resulting generated sigma-field by $\sigma(\Omega^{(n)})$. Suppose $X$ is an $\mathbb{R}^d$-valued random variable and $f:\mathbb{R}^d\to\mathbb{R}$ is a convex function.

Is the following true? $$ \mathbb{E}[f(\mathbb{E}[X:\sigma(\Omega^{(n)})])] \leq \mathbb{E}[f(\mathbb{E}[X:\sigma(\Omega^{(n+1)})])] $$

I've done some numerical experiments and it seems to suggest the above is true. However, I can't seem to find any reference on the internet.

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1 Answer 1

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It seems that simple counterexamples abound. You might want to check the following: Assume that $\{A,B,C,D\}$ is a partition of $\Omega$ with $$P(A)=P(D)=\tfrac13\qquad P(B)=P(C)=\tfrac16$$ and consider the equipartitions $$\Omega^{(2)}=\{A\cup B,C\cup D\}\qquad\Omega^{(3)}=\{A,B\cup C,D\}$$ and the random variable $$X=3\mathbf 1_B-9\mathbf 1_C$$ Then $$E(X\mid\sigma(\Omega^{(2)}))=\mathbf 1_{A\cup B}-3\mathbf 1_{C\cup D}\qquad E(X\mid\sigma(\Omega^{(3)}))=-3\mathbf 1_{B\cup C}$$ hence $$E(E(X\mid\sigma(\Omega^{(2)}))^2)=5\qquad E(E(X\mid\sigma(\Omega^{(3)}))^2)=3$$

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  • $\begingroup$ In my experiments, the expected value $E[X] \geq 0$. Do you think this would make a difference? $\endgroup$ Oct 6, 2017 at 9:58
  • $\begingroup$ Nope: consider $X_x=X+x$ and $f_x(t)=(t-x)^2$ instead of $X$ and $f(t)=t^2$. $\endgroup$
    – Did
    Oct 6, 2017 at 10:39

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