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For two $\sigma$-fields, $\mathcal{A}$ and $\mathcal{B},$ we have the notion of HGR maximal correlation

$$\rho(\mathcal{A},\mathcal{B}) = \sup \frac{Efg-Ef.Eg}{\sqrt{\mathsf{Var}(f).\mathsf{Var}(g)}}$$

where the supremum is over $f,g$ being $L^2$ functions measurable with respect to $\mathcal{A}$ and $\mathcal{B}$ respectively.

We also have the notion of correlation

$$\delta(\mathcal{A},\mathcal{B}) = \sup_{A\in\mathcal{A}, B\in\mathcal{B}} \frac{P(A\cap B)-P(A)P(B)}{\sqrt{P(A)P(A^c)P(B)P(B^c)}},$$

that is, the supremum over $f,g$ being indicator functions measurable with respect to $\mathcal{A}$ and $\mathcal{B}$ respectively.

Is it true that these two measures of correlation are identical when $\mathcal{A}$ and $\mathcal{B}$ both have finite number of sets?

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A good reference for mixing condition is Bradley's book or paper .

The notion of $\lambda$-mixing is introduced, and defined by $$\lambda(\mathcal A,\mathcal B):=\sup_{A\in\mathcal A,B\in\mathcal B}\frac{|\mu(A\cap B)-\mu(A)\mu(B)|}{\sqrt{\mu(A)\mu(B)}}\leqslant \delta (\mathcal A,\mathcal B).$$

He proves that $$\lambda(\mathcal A,\mathcal B)\leqslant\rho(\mathcal A,\mathcal B)\leqslant 52\lambda(\mathcal A,\mathcal B)\log(1-\lambda(\mathcal A,\mathcal B)),$$ so $\delta$ and $\rho$-mixing coefficients are linked in this way (and it's sufficient for the applications, I think).

Probably there is not equality in the case of finite $\sigma$-algebras (except when they are generated by two elements) because by an approximation argument this would extend to separable random fields (that is generated by countably many random variables).

  • Bradley, R.M., Introduction to strong mixing conditions, volume 1.
  • Bradley, R.M., Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
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