Multivariate Probability Distribution with Linear Conditional Expectation I want to know what probability distribution has the linearity property of the conditional expectation.
To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$.
Then, if $[v_1,\;v_2,\;v_3]$ follows a joint normal distribution, we can show that $\mathbb{E}[v_1|v_2,\;v_3]$ is linear in $v_2$ and $v_3$.
That is, $\mathbb{E}[v_1|v_2,\;v_3]=\rho'[v_2,\;v_3]'$, where $\rho$ is a $2\times1$ constant vector consists of covariance of $[v_1,\;v_2,\;v_3]$.
Here, according to a textbook, there are diverse probability distributions that have the same linearity of the conditional expectation.
But, I am not sure what distributions also have that property.
Could you give me some examples (with proof if possible).

Multivariate Elliptical distributions deals with linearity in conditional expectation. You can think about this family as a generalization of Normal distribution,  t-Student is another notable example of this family.
