On the rate of convergence of random polynomials of degree 3 (Q1600651)

Let \(x_1,x_2,\dots,x_n\) be iid random variables. Let \(P_3=P_3(x_1,\dots,x_n)\) denote a sequence of homogeneous symmetric random polynomials of degree three and \(Z_3(x)=n^{-3/2}P_3(x)\). Let \(z=a_1N^3+a_2N\), where \(N\) is the standard normal law. The author obtains an estimate of the order of convergence rate as \(n^{-1/2}\) in the Lévy metric. It is really an improved estimate.