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If $X$ and $Y$ are two random variables, then the law of total variance allows us to calculate the first moment of $\text{Var}(X|Y)$ by $$E[\text{Var}(X|Y)] = \text{Var}(X) - \text{Var}(E[X|Y]).$$ In particular we have that $E[\text{Var}(X|Y)] < \text{Var}(X)$.

I'm wondering if similar relations may hold for higher moments of $\text{Var}(X|Y)$. For example can we say something about $E[\text{Var}(X|Y)^2]$ compared to $\text{Var}(X)^2$?

I will just comment that for any $n\in \mathbb{N}$ we have the following $$\text{Var}(X|Y)^n \leq E[X^2|Y]^n$$ and so $$E[\text{Var}(X|Y)^n]\leq E[X^{2n}].$$

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