What is the Cauchy meta distribution? I overhead a professor speak about the Cauchy meta-distribution, but I am unable to find anything about it on the web. My question is what is the Cauchy meta distribution and what is the theory behind it?
 A: A meta distribution is one with normal marginals but a joint copula based on a stronger tail-dependence. So a $t_\nu$ meta distribution would be obtained by generating a multivariate-t with $\nu$ d.f. and then transforming all the marginals to normals.
See, for example, page 5 of this document.
Of course, a Cauchy is just a $t_1$.
There's a picture of a bivariate sample from a Cauchy meta distribution on p7 of the same document, and a bunch of references further on.
You might want to work on your google-fu - I found that in very short order and I'd never heard of it before (though I was familiar with the concept once I saw what it was).
Here's a paper I found with the same search
Some R code (not the fastest way to do it):
library(MASS)  #for mvrnorm
n <- 10000
cauchysamp <- mvrnorm(n,mu=c(0,0),Sigma=matrix(c(1,.9,.9,1),nr=2))/rnorm(n)
a <- qnorm(pt(cauchysamp,df=1))
plot(a,cex=.5)

gives:

which looks like the plot on page 7 I linked to before. Note that a[,1] and a[,2] are each univariate normal. In the code, immediately before it calculates the qnorm (after the pt), you have a Cauchy copula.
