# Joint distribution of independent t-distributed random variables

The multivariate t distribution seems to be defined as a "ratio" of a vector of normal random variables and a single gamma (or chi-squared) random variable (independent from the vector of normal).

Thus, even if the normal rvs are independent (i.e. generated from a multivariate normal with a diagonal covariance matrix) this definition seems to, nevertheless, imply that the random variables within a vector draw from a multivariate t are not independent since the same gamma rv is used to scale all normal rvs.

What is then a joint distribution of independent t-distributed rvs? Also, why in the definition of the multivariate t, we use only one gamma rv to construct a vector of t rvs from a vector of normal, instead of using a vector of gamma rvs so that each normal rv is scaled by a different gamma rv?

Let $\mathbf{X} \sim N_p(\mu, \Sigma)$ be a multivariate Normal random variable, and $Y \sim \chi^2_{\nu}$ be a $\chi^2$ random variable with $\nu$ degrees of freedom. Then the random variable $\mathbf{X}/\sqrt{Y/\nu}$ follows a Multivariate $t$ distribution with $\nu$ degrees of freedom. The pdf of this distribution is

$$f(\mathbf{x}) = \dfrac{\Gamma((\nu+p)/2)}{\Gamma(\nu/2) \nu^{p/2} \pi^{p/2} |\Sigma|^{1/2} } \left[1 + \dfrac{(\mathbf{x} - \mu)^T \Sigma^{-1} (\mathbf{x}-\mu)}{\nu} \right]^{-(v+p)/2} \,.$$

When $\Sigma$ is a diagonal matrix, the components of $\mathbf{X}$ are independent normal, but note that the pdf of the resulting multivariate $t$ distribution does not decompose into the product of the marginals, since the pdf is:

$$f(\mathbf{x}) = \dfrac{\Gamma((\nu+p)/2)}{\Gamma(\nu/2) \nu^{p/2} \pi^{p/2} |\Sigma|^{1/2} } \left[1 + \dfrac{1}{\nu}\sum_{i=1}^{p}\dfrac{(\mathbf{x} - \mu)^2 }{\sigma^2_i} \right]^{-(v+p)/2} \,.$$

Thus, independence of the underlying normal random variables, does not imply independence of the resulting $t$ variables. Also not that, the since the $\Sigma$ matrix is a diagonal matrix, the $t$ marginal variables are uncorrelated (but not independent).

So the joint distribution of uncorrelated $t$-random variables is above. As for the joint distribution of uncorrelated and independent $t$ random variables, that would just be the product of marginals.

As for using separate $\chi^2$ distributions, for a diagonal $\Sigma$ matrix that should correspond to independent univariate $t$ distributions. I am not quite sure what happens for a non-diagonal $\Sigma$ in this case.

• Thank you for your answer. However, the first 4 paragraphs of your answer are already described in the OP; and the question is precisely about a pdf of an rv constructed using a multivariate normal and a vector of (independent) χ2, to which your reply is "I am not quite sure what happens for a non-diagonal Σ in this case." So, I cannot accept it as an answer, although I will up-vote it. Thanks again. – Confounded Mar 13 '18 at 16:11
• @Confounded I agree I haven't completely answered your question. I will argue that the first 4 paragraphs of the answer are not entirely described in the question. The purpose of the first 4 paragraphs is to explain the delicate difference here between dependence and correlations. This is not described in the original question. – Greenparker Mar 13 '18 at 16:16