Optimal rates of convergence of second-degree polynomials in several metrics (Q876855)

Let \(X_1, X_2,\dots\) be a sequence of zero mean, independent and identically distributed random variables with finite fourth moments. Assume the variance of \(X_1\) is 1. Let \(Y_k = X_k^2-1\), \(T_1 = n^{-1}(Y_1+\dots+Y_n)\) and \(T_2=n^{-1/2}(X_1+ \dots + X_n)\). The paper considers the rate at which \(T=T_2^2 - T_1 -1\) converges to a mean adjusted \(\chi^2_1\) distribution as measured by different metrics. In particular the optimal rate of convergence is obtained for the Kolmogorov distance and for the Lévy distance in the sense that the rates are achieved in the case where the \(X_i\) have a standard normal distribution.