On the equidistribution of polynomials of normal random variables (Q1897899)

Let \(N_1, \dots, N_l\) be random variables \(N(0,1)\) with correlation coefficients \(\rho_{ij}\). Let \(a_2, \dots, a_k\), \(b_1, \dots, b_l\), \(a_2', \dots, a_k'\), \(b_1', \dots, b_l'\) be some coefficients and \(X {\buildrel \text{d} \over =} Y\) denote the equidistribution of random variables \(X\) and \(Y\). A necessary and sufficient condition for \[ a_k N^k_1 + \cdots + a_2 N^2_1 + b_1 N_1 + \cdots + b_l N_l {\buildrel \text{d} \over =} a_k' N_1^k + \cdots + a_2' N_1^2 + b_1' N_1 + \cdots + b_l' N_l \] is proved by using properties of coefficients and of the \(\rho_{ij}\). This theorem is important for the study of the convergence (of the limit behaviour) of certain normed random polynomials.