Doukhan, Paul Fonctions d’Hermite et statistiques des processus melangeants. (Hermite functions and statistics of mixing processes). (French) Zbl 0613.62050 Cah. Cent. Étud. Rech. Opér. 28, 99-115 (1986). In this paper orthogonal series estimates based on Hermite functions for probability densities and regression functions and their derivatives are investigated. Several measures of deviation, e.g. the MISE on \({\mathbb{R}}\) or the expected uniform deviation on a compact interval are explored. In case of independent samples e.g. S. C. Schwartz [Ann. Math. Stat. 38, 1261-1265 (1967; Zbl 0157.479)], G. G. Walter [Ann. Stat. 5, 1258-1264 (1977; Zbl 0375.62041)] or W. Greblicki and M. Pawlak [J. Multivariate Anal. 15, 174-182 (1984; Zbl 0544.62043)] have evaluated the MISE, whereas in this paper stationary \(\phi\)- and strongly mixing samples are admitted too. Furthermore e.g. a central limit theorem for the \(L_ 2\)-deviation of the estimates, the behavior of a kernel estimate related to the Hermite-system (Mehler kernel) and a weak invariance principle for empirical distributions with an index space containing functions with small Fourier coefficients are presented as well. Reviewer: U.Stadtmüller MSC: 62G05 Nonparametric estimation 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 60F05 Central limit and other weak theorems Keywords:quadratic deviation; quadratic uniform risk; density estimation; asymptotic normality; phi-mixing; orthogonal series estimates; Hermite functions; measures of deviation; MISE; expected uniform deviation; strongly mixing; central limit theorem; Mehler kernel; weak invariance principle; empirical distributions Citations:Zbl 0157.479; Zbl 0375.62041; Zbl 0544.62043 PDFBibTeX XMLCite \textit{P. Doukhan}, Cah. Cent. Étud. Rech. Opér. 28, 99--115 (1986; Zbl 0613.62050)