Is there any other multivariate distribution $f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)$ that zero covariance between $X_i$ and $X_j$ implies independence between $X_i$ and $X_j$ besides multivariate normal distribution? For example, if multivariate t distribution $f(t_1, t_2)$ has $Cov(T_1, T_2)=0$ and it implies that $T_1$ and $T_2$ are independent, then multivariate t distribution counts as one such distribution as multivariate normal.
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$\begingroup$ Sure, because you use the word "distribution" in the sense of a family of distributions. Any family whose members are all multivariate distributions of independent variables trivially has this property. Perhaps you intended to mean "distribution" in a more restrictive sense--but what would that be? $\endgroup$– whuber ♦Commented Oct 21, 2019 at 19:11
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$\begingroup$ If you know that they are independent, then you don't need to prove independence. My question is, if we start from the joint pdf of some multivariate distribution and zero variance, under what kind of distribution can we reach to independence? multivariate normal is a common one, but what else? Multivariate t? $\endgroup$– Han ChenCommented Oct 21, 2019 at 19:22
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$\begingroup$ The general answer to your general question is any family of distributions in which the ones with zero covariance are also independent. Your reference to "prove" is a little mystifying, because you are asking only for characterizations. For the reasons I am giving, I consider this question to be overly broad: is there any way you can make it more specific? $\endgroup$– whuber ♦Commented Oct 21, 2019 at 21:23
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$\begingroup$ Thanks! I think I've found an example, multivariate Bernoulli distribution, which uncorrelatedness implies independence just like multivariate Normal distribution. $\endgroup$– Han ChenCommented Oct 22, 2019 at 16:54
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$\begingroup$ It depends on which family of multivariate Bernoulli distributions you have in mind. There are families where uncorrelatedness does not imply independence. For instance, they might include the example in this answer where each pair of three Bernoulli variables is independent (and therefore uncorrelated) but the three variables are not independent. $\endgroup$– whuber ♦Commented Oct 22, 2019 at 17:10
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