Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be a continuous linear map.
Let $\mathbb P$ be a probability measure on $(X, \mathcal B(X))$. The covariance inner product is defined on $X^*$ in the usual way.
We say that a functional $\varphi \in X^*$ is $f$-uncorrelated (with respect to $\mathbb P$) if $\varphi$ and $f^* \psi$ are uncorrelated for all $\psi \in Y^*$. i.e., $\operatorname{cov}_\mathbb P[\varphi|f^* \psi] = 0$.
We say that $\varphi$ is $f$-independent (with respect to $\mathbb P$) if the $\sigma$-algebras $\sigma(\varphi)$ and $\sigma(f)$ are independent (with respect to $\mathbb P$).
We say that $\mathbb P$ satisfies the "uncorrelated implies independent" (UII) property (with respect to $f$) if $$\mbox{$\varphi$ is $f$-uncorrelated implies that $\varphi$ is $f$-independent}$$for all functionals $\varphi \in X^*$.
It is not hard to see that Gaussian measures satisfy the UII property, and in fact this is the impetus for the general definition.
Question: What other classes of measures satisfy the UII property for all continuous linear maps?
Note that we have made no assumptions about the support of $\mathbb P$. A simple non-Gaussian example is the case of two coupled binomial random variables. However, this limits us to 2 degrees of freedom, whereas the Gaussian allows for infinitely many degrees of freedom.
I am hoping there are some interesting examples of non-Gaussian UII measures with infinitely many degrees of freedom. If they do not exist, I will be very interested to see the proof.
