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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 $\mathbb{P} \left[ \det\left( \mathbf{A}\mathbf{A}^H \right) < b \right]$?

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    $\begingroup$ Here's a characterization of the distribution in terms of $\chi^2$: mathoverflow.net/a/268156/4600 $\endgroup$ Jan 7 at 8:04
  • $\begingroup$ 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. $\endgroup$ Jan 7 at 8:16
  • $\begingroup$ 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. $\endgroup$ Jan 7 at 13:32

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