Fourth-moment bound for a centered complex quadratic form

First fix a deterministic complex matrix $A$. Centering the diagonal terms gives ${\xi }^{\ast }A\xi -\mathrm{tr}A={\sum }_i{A}_{ii}(\mathrm{\mid }{\xi }_i{\mathrm{\mid }}^2-1)+{\sum }_{i\mathrm{\ne }j}\overline{{\xi }_i}{A}_{ij}{\xi }_j\mathrm{.}$ In the squared modulus, independence and $\mathbb{E}{\xi }_i=0$ make diagonal and off-diagonal parts uncorrelated; distinct diagonal terms are uncorrelated; and two off-diagonal terms have nonzero expectation only when their ordered index pairs agree or are reversed. Expanding those remaining terms yields $\mathbb{E}\mathrm{\mid }{\xi }^{\ast }A\xi -\mathrm{tr}A{\mathrm{\mid }}^2=({\mu }_4-1){\sum }_i\mathrm{\mid }{A}_{ii}{\mathrm{\mid }}^2+{\sum }_{i\mathrm{\ne }j}\mathrm{\mid }{A}_{ij}{\mathrm{\mid }}^2+\mathrm{\mid }\tau {\mathrm{\mid }}^2{\sum }_{i\mathrm{\ne }j}{A}_{ij}\overline{A_{ji}}\mathrm{.}$ Here ${\mu }_4-1\ge 0$ by Cauchy–Schwarz, and $\mathrm{\mid }\tau \mathrm{\mid }\le \mathbb{E}\mathrm{\mid }{\xi }_1{\mathrm{\mid }}^2=1$. Also $\mid {\sum }_{i\mathrm{\ne }j}{A}_{ij}\overline{A_{ji}}\mid \le {\sum }_{i\mathrm{\ne }j}\mathrm{\mid }{A}_{ij}\mathrm{\mid }\mathrm{\mid }{A}_{ji}\mathrm{\mid }\le {\sum }_{i\mathrm{\ne }j}\mathrm{\mid }{A}_{ij}{\mathrm{\mid }}^2\mathrm{.}$ Thus the deterministic quadratic-form variance is at most $C_{{\mu }_4,\tau }\mathrm{\parallel }A{\mathrm{\parallel }}_F^2$.

By [F2], $(\xi ,A)$ has product joint law. The integrand is integrable because $\mathrm{\mid }{\xi }^{\ast }A\xi -\mathrm{tr}A{\mathrm{\mid }}^2\le 2\mathrm{\parallel }A{\mathrm{\parallel }}_F^2\mathrm{\parallel }\xi {\mathrm{\parallel }}_2^4+2d\mathrm{\parallel }A{\mathrm{\parallel }}_F^2,$ and $\mathbb{E}\mathrm{\parallel }\xi {\mathrm{\parallel }}_2^4<\mathrm{\infty }$ in finite dimension. Integrate the deterministic inequality from step 1.1 over the law of $A$; [F3] justifies the order of integration and gives the stated bound.