ИСТИНА

New version of the central limit theorem for an array of the row-wise conditionally independent (with respect to certain sigma-algebras) random variables is established. This result can be viewed as an analog of the classical Lindeberg - Feller theorem known for families of independent random variables. In recent paper [3] the authors studied the a.s. convergence of the conditional characteristic functions of appropriately normalized partial sums of conditionally independent (with respect to specified sigma-algebra) summands. In contrast to [3] we consider more general scheme of arrays and use the convergence in probability. The application of the obtained conditional central limit theorem is provided. Namely, we extend the main result of [2] concerning the asymptotic normality of the second moment of the regression function estimate. The analysis of such estimates behavior is related to the feature selection problem (see, e.g., [1]). References [1] V.Bolon-Canedo, N.Sanchez-Marono and A.Alonso-Betanzos. Feature Selection for High Dimensional Data. Springer, Cham, 2015. [2] L.Gyorfi and H.Walk. On the asymptotic normality of an estimate of a regression functional. J. of Machine Learning Research, 2015, v.16, 1863- 1877. [3] D-M. Yuan, L-R. Wei and L. Lei. Conditional central limit theorems for a sequence of conditional independent random variables. J. Korean Math. Soc., 2014, v. 51, no. 1, p. 1-15.