# Basics of Multivariate Student's t distribution

I am a fan of the Student's t distribution -- it is almost as easy to handle as the normal distribution but has an additional flexibility with respect to the heaviness of the tails. In fact, the normal distribution is a special case and so is the Cauchy distribution.

The density of a (non-degenerate) multivariate normal distribution with zero mean is given by

$\frac{1}{\sqrt{(2\pi)^n |\Sigma|}} \exp \Big( -\frac{1}{2} x^\top \Sigma^{-1} x \Big),$

where $$\Sigma$$ is a (positive-semidefinite) $$n\times n$$ covariance matrix.

For comparison, the density of the multivariate student t distribution with zero mean is

$\frac{\Gamma[(\nu + n)/2]}{\Gamma(\nu/2) \sqrt{v^{n} \pi^{n} |\Sigma|}} \Big[ 1 + \frac{1}{\nu} x^\top \Sigma^{-1} x \Big]^{-(\nu + n)/2}.$

Compare this with the univariate version of the t density:

$\frac{\Gamma{(\nu +1)/2}}{\Gamma(\nu/2)\sqrt{\nu\pi}} \Big[ 1 + \frac{x^2}{2} \Big]^{-(\nu+1)/2}.$

In both cases, $$\nu>2$$ is the degrees of freedom parameter.

A standard way to construct a random variable $$Z$$ with t-distribution is to use a normal random variable $$X$$ and an independent chi-square random variable $$Y$$ and set

$Z = \frac{X}{\sqrt{Y/\nu}}.$

This formula works for both the univariate and the multivariate case and can be used to draw samples from the t distribution using independent samples of $$X$$ and $$Y$$. Also, we get

$\text{Cov} Z = \frac{\nu}{\nu-2} \text{Cov} X.$

This is because the univariate $$\chi^2$$ distribution with $$\nu$$ degrees of freedom is really a $$\text{Gamma}(\frac{\nu}{2}, \frac{1}{2})$$ distribution. This can be seen by comparing a $$\chi^2$$ density with the $$\text{Gamma}(\alpha, \beta)$$ density

$p_{\alpha,\beta}(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} 1_{[0, \infty)}(x).$

The parameters $$\alpha$$ and $$\beta$$ are both positive. Moreover, $$1/Y$$ is inverse-gamma distributed and thus $$\mathbb E \frac{1}{Y} = \frac{1}{\nu -2}.$$