Normalization in the estimation of the quality of statistical procedures (Q1363884)

It was established in the theory of random matrices of increasing dimension [see \textit{V. A. Marchenko} and \textit{L. A. Pastur}, Mat. Sb., n. Ser. 72(114), 507-536 (1967; Zbl 0152.16101); \textit{V. L. Girko}, ``Random matrices'' (1975; Zbl 0344.60001) and ``Spectral theory of random matrices'' (1988; Zbl 0656.15012); and the author, Usp. Mat. Nauk 40, No. 2(242), 197-198 (1985; Zbl 0572.60043)], under certain weak conditions on the degree of dependence between variables, that spectral functions of random matrices can have finite biases and decreasing variances, and yet tend to limits of functions, that depend only on the first two moments of variables. One can deduce from this fact that functions of the type of functions of quality of statistical procedures can admit a reliable estimation that is based on the hypothesis of the normality of observations. We will study classes of functionals that have this property and establish required conditions.