Conditional Central Limit Theorem

Abstract

The arrays with rows consisting of conditionally independent random variables with respect to certain $\sigma$-algebras are studied. An analogue of the Lindeberg--Feller theorem known for systems of independent random variables is established. This result is based on the theorem proved by Yuan, Wei, and Lei in [J. Korean Math. Soc., 51 (2014), pp. 1--15], where the authors considered a sequence of random variables conditionally independent with respect to a given $\sigma$-algebra. They were interested in a.s. convergence, whereas our version of the Lindeberg condition in a weak form (involving convergence in probability) is less restrictive. An application of the indicated new result for arrays provides an extension of conditions for asymptotic normality of the estimates of the regression function second moment obtained in a recent paper by Gyorfi and Walk [J. Mach. Learn. Res., 16 (2015), pp. 1863--1877].