A central limit theorem for Hermitian polynomials of independent Gaussian variables (Q2640982)

Let \(H(x_ 1,...,x_ r)\) be a Hermitian polynomial, \[ H(x_ 1,...,x_ r)=(\prod_{1\leq i\leq j\leq r}(\nabla_ i,\nabla_ j)^{k_{ij}}\phi (x_ 1,...,x_ r))\phi^{-1}(x_ 1,...,x_ r), \] where \(\nabla_ i=\partial /\partial x_ i\), \(k_{ij}\in Z_+\), \(\phi (x_ 1,...,x_ r)=\exp \{-2^{-1}\sum^{r}_{\ell =1}| x_{\ell}|^ 2\}\). The purpose of this paper is to study the problem of asymptotic behaviour of the \(\eta_ n=\sum_{1\leq i_ 1\leq...<i_ r\leq n}H(\xi_{i_ 1},...,\xi_{i_ r}),\) where \(\xi_ 1,\xi_ 2,...,\xi_ n\) are independent Gaussian vectors in \(R^ m\) with zero mean and unit correlation operator. It is shown that with regard to asymptotic normality of \(\eta_ n/D\eta_ n\) if n,m\(\to \infty\), it is necessary and sufficient that \(\sum_{1\leq i<j\leq r}k_{ij}>0\).