Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Magic! I have never seen anything like this before. This calculus feels like real magic. Are there any books on the usage of this technique? Especially on random determinants.
Apart of combinatorial proof discussed above, one can prove the fourth moment formula by expanding the determinant wrt one row and then find recurrence relations
Does it imply that since the density of $Z$ depends actually on any strictly monotone function $ g(\left \| Z \right \|_2)$, so the pair of random variables $Z/g(\left \| Z \right \|_2)$ and $g(\left \| Z \right \|_2)$ are independent as well?
@Stanley After more than a year of (from time to time) trying to tackle the poblem, I have finally concluded it cannot be done this way. It is, however, rather strange. Nyquist (1954) derived an almost general formula for E[det^4 A]. However, his entries are symetrical random variables (the case of all $\pm 1$ matrices belongs here! you can even assign some probability $p$ that the entries are zero). Somehow, however, there is no such formula for the case when the A's entries are general iid random variables.
I have to apologize for adding and answer so late, I was in the meantime studying at uni so I forgot where this page was until now, the answer is I think according to same approach that the integral diverges to infinity. The terms will not cancel out.
However, this could be a nice proof of $\int_{0}^{\pi/2}\sin^2 x\,\mathrm{d}x = \int_{0}^{\pi/2}\cos^2 x\,\mathrm{d}x = \frac{1}{2}\left(\frac{\pi}{2}\cdot 1\right)$
