2.5 Distribution of the Determinant of a Complex-Valued Sample Correlation Matrix

In this post, we look at the distribution of the determinant of the sample correlation matrix of the realizations of a complex-valued Gaussian random vector. The distribution for real-valued Gaussian random vector was developed in [18], and we largely follow the developed framework. Thanks to Prashant Karunakaran for bringing this problem and the many applications to my attention in late 2017/early 2018.

Let $𝒙$ be a Gaussian random vector of length $p$ with mean $𝝁\in {ℂ}^p$ and covariance $𝚺\in {ℂ}^{p\times p}.$ Let ${𝒙}^{(1)},{𝒙}^{(2)},\mathrm{\dots },{𝒙}^{(n)}$ denote $n$ realizations, $n\ge p,$ of $𝒙.$ In the terminology of [18], the adjusted sample covariance matrix is given by

where $\overline{𝒙}$ is the sample mean given by

Note that the adjusted sample covariance matrix is positive semi-definite.

The correlation matrix $𝑹$ is defined as:

where $𝑫=\mathrm{Diag}(𝑺)$ is a diagonal matrix with the diagonal elements of $𝑺$ on the main diagonal. Hence, $𝑹$ has unit diagonal elements and is independent of the variance of the elements of $𝒙.$

Now, for real-valued $𝒙,$ the determinant of $𝑹,$ denoted by $|𝑹|,$ is shown in [18, Theorem 2] to be a product of $p-1$ Beta-distributed scalar variables $\mathrm{Beta}(\frac{n-i}{2},\frac{i-1}{2}),$ $i=1,\mathrm{\dots },p-1.$ The density of the product can be given in terms of the $\mathrm{MeijerG}$ function as follows [18, Theorem 2]:

Analogously, for complex-valued $𝒙,$ $|𝑹|$ is a product of $p-1$ Beta-distributed scalar variables $\mathrm{Beta}(n-i,i),$ $i=1,\mathrm{\dots },p-1.$ The density of the product can now be given in terms of the $\mathrm{MeijerG}$ function, in a straightforward manner, as follows.

In the following, a Mathematica program for numerical simulation and the corresponding output are provided.

The following figure shows that the numerical and analytical results match perfectly for the example case $n=4,k=6.$