# The CDF of determinant of wishart distribution

Assuming $$b>0$$, $$\mathbf{A} \in \mathbb{C}^{m\times n}$$ with $$m\leq n$$ and each element of $$\mathbf{A}$$ is i.i.d. $$\mathcal{CN}(0,1)$$ distributed, how to obtain $$\mathbb{P} \left[ \det\left( \mathbf{A}\mathbf{A}^H \right) < b \right]$$?

• Here's a characterization of the distribution in terms of $\chi^2$: mathoverflow.net/a/268156/4600 Jan 7 at 8:04
• It means that the determinant is distributed as the product of independent random variables with a chi-squared distribution, but I don't know that how to obtain the CDF of the product of independent random variables with a chi-squared distribution. Jan 7 at 8:16
• the linked answer gives the asymptotics of the distribution for large $m$ and $n$; if $m$ and $n$ are small you can compute the CDF directly from the $\chi^2$ distributions; if $n,m$ are neither small nor large then there is no practical way to obtain a closed-form expression for the CDF. Jan 7 at 13:32