On a universal strong law of large numbers for conditional expectations

Abstract

A number of generalizations of the Kolmogorov strong law of large numbers are known including convex combinations of random variables (rvs) with random coefficients. In the case of pairs of i.i.d. rvs $(X_1,Y_1),...,(X_n,Y_n)$, with $\mu$ being the probability distribution of the $x$s, the averages of the $Y$s for which the accompanying $X$s are in a vicinity of a given point $x$ may converge with probability 1 (w.p. 1) and for $\mu$-almost everywhere ($\mu$ a.e.) $x$ to conditional expectation $r\left(x\right)=E\left(Y\right|X=x)$. We consider the Nadaraya-Watson estimator of $E\left(Y\right|X=x)$ where the vicinities of $x$ are determined by window widths $h_n$. Its convergence towards $r\left(x\right)$ w.p. 1 and for $\mu$ a.e. $x$ under the condition $E\left|Y\right|<\mathrm{\infty }$ is called a strong law of large numbers for conditional expectations (SLLNCE). If no other assumptions on $\mu$ except that implied by $E\left|Y\right|<\mathrm{\infty }$ are required then the SLLNCE is called universal. In the present paper we investigate the minimal assumptions for the SLLNCE and for the universal SLLNCE. We improve the best-known results in this direction.