I need to sample from a multivariate t-distribution with less than 2 degrees of freedom.

This article on Wikipedia (https://en.wikipedia.org/wiki/Multivariate_t-distribution#Definition) shows that a multivariate t-distribution with ν\nu degrees of freedom can be defined as follows:

X=μ+√WAZ∼tν(μ,Σ)X = \mu + \sqrt{W}AZ \sim t_\nu(\mu, \Sigma)

where W=ν/χ2νW=\nu/\chi_{\nu}^2 and χ2ν\chi_{\nu}^2 is a random variable following a chi-squared distribution with ν>0\nu>0 degrees of freedom, Z∼N(0,Σ)Z\sim N(0,\Sigma) is a normal random vector with independent components. Note that χ2ν\chi_{\nu}^2 is independent of ZZ.

But since ν<2\nu<2, the covariance matrix of XX (i.e. Σ\Sigma) does not exist, and therefore I cannot sample ZZ. Is there another method I should be using for sampling? How can I sample a t distribution with less than 2 degrees of freedom? ================= Most simulations are based on the LLN, which requires finite variance. What attribute of the multivariate t are you trying to simulate? Is there an analogue that can be tested with the univariate t so you can check stability at least in that special case? // If you decide the project is valid, then perhaps use Metropolis-Hastings; I think you will have trouble finding a valid majorizing function for simpler acceptance-rejection methods. – BruceET Oct 20 at 22:34 First it depends on which multivariate generalization you are referring to. Once you have chosen one, and if you know its joint pdf, I think you can use acceptance rejection. – BGM 2 days ago ================= =================