Correlation matrix for multivariate Cauchy distribution

Question

I have found an equation for the entropy of a $p$-variate Cauchy distribution here [page 70]:

$H(X,R) = \frac{1}{2}\log(\det(R))+f(p)\,,$

where $X=(X_1,X_2,\dots,X_p)$ is vector of random variables having a $p$-variate Cauchy distribution, and $R$ is the correlation matrix of $X$, and $f$ is some function.

I have problem understanding how one can define a correlation matrix for a multivariate Cauchy distribution. Normally, one would define the a correlation $R_{ij}$ via $Cov(X_i,X_j)$ and the variances of $X_i$ and $X_j$. But for the Cauchy distribution, the variances are undefined. So how would I define correlation for the multivariate Cauchy distribution?

Answer 1

Answered in comments:

That's not actually a correlation matrix, it's a scale matrix. It so happens that for the multivariate Normal distribution with all the variances equal to 1, the scale matrix is a correlation matrix, but otherwise not. So ... it will exist even for a Cauchy distribution. – jbowman