A central limit theorem for Hermitian polynomials of independent Gaussian variables

Abstract

The conditions of asymptotic normality of the variables$\eta _n = _{1 \leqslant i_j< ...< i, \leqslant 11}^{ \Sigma H(\delta _{lj} ,...,\delta _{ir} )}$ are studied for n→∞ and m→∞, with H(x₁, ..., x_r) denoting Hermitian polynomials in (R^m)^r, and the ξ₁, ..., ξ_n being independent Gaussian vectors in X=R^m with a zero mean and a unit correlation operator.