Suppose $A$ is a real $N \times P$ matrix, $P \geq N$, with entries drawn independently according to $A_{ij} \sim \mathcal{N}(0,1)$. Then $W = A \, A^\top$ is a member of the real Wishart ensemble. What is the distribution of $\det W$? I'm particularly interested in the large $N$ behaviour. There might be a qualitative distinction between cases when $N / P(N) \rightarrow c$, with $c = 1$ or $c < 1$. In that case I'd be more interested in the former. Thanks!
Real Wishart Matrix – Determinant Calculation
determinantsrandom matrices
Best Answer
The Distribution of the Determinant of a Complex Wishart Distributed Matrix proves that the determinant is distributed as the product of independent random variables with a chi-squared distribution, $${\rm det}\,W\simeq\chi^2_P\chi^2_{P-1}\cdots\chi^2_{P-N+1}.$$ (The title of the paper refers to the complex case, but the real ensemble is also considered towards the end.)
Results for the large-$N$ asymptotics can be found here. If we take $N,P\rightarrow\infty$ at fixed ratio $c=N/P<1$, the asymptotics is $$\frac{\log{\rm det}\,(W/P)-\sum_{k=1}^N\log(1-k/P)}{\sqrt{-2\log(1-c)}}\rightarrow z,$$ with $z$ normally distributed. For $c\rightarrow 0$ this means that $(2c)^{-1/2}\log{\rm det}\,(W/P)$ tends to a normal distribution. At the other extreme, for $c\rightarrow 1$ one has $$\frac{\log{\rm det}\,(W/P)+P+\tfrac{1}{2}\log(P/2\pi)}{\sqrt{2\log P}}\rightarrow z,$$