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.



Measures which exhibit the "uncorrelated implies independent" property
Associated variables satisfy the UII property – see e.g. Newman, Charles M. "Asymptotic independence and limit theorems for positively and negatively dependent random variables." Lecture Notes-Monograph Series (1984): 129. Available [here](http://books.google.pl/books?id=neFsjyQrCEwC&pg=PA129&lpg=PA129&dq=%22%20uncorrelated%20implies%20%22%20independent%20probability&source=bl&ots=2tS5PDGxbh&sig=RpX2N7YtWoRmXJ0c2S_Nr87bLuE&hl=pl&sa=X&ei=Jo06U6qoI8erhAeA0YCIDg&redir_esc=y#v=onepage&q=%22%20uncorrelated%20implies%20%22%20independent%20probability&f=false)


Thus e.g. we know that random variables with a multivariate extreme value distribution satisfy the property - [Marshall, Albert W., and Ingram Olkin. "Domains of attraction of multivariate extreme value distributions." The Annals of Probability 11.1 (1983): 168-177](https://statistics.stanford.edu/sites/default/files/OLK%20NSF%20171.pdf).


