Normal Variance Mixtures and the Multivariate t Distribution

The multivariate normal is rejected by Mardia's tests for daily and weekly financial returns. The fix cannot be "more parameters" — what is needed is a different mechanism for generating heavy tails and joint extremes. Normal variance mixtures supply this: the idea is conceptually simple — randomise the volatility of a multivariate normal by multiplying it by a positive scalar random variable $W$. The resulting distribution retains the multivariate normal's linear-algebra tractability (linear combinations remain in the same family, conditional distributions are characterised, Cholesky simulation still works) while generating heavy marginal tails and positive tail dependence. The most important special case is the multivariate $t$ distribution, which arises from an inverse-Gamma mixing variable and has tail index $\nu$ — the same parameter that governs the marginal tail behaviour as in the univariate case studied in the EVT topic.

Learning objectives. By the end of this note you should be able to: