Strong law for randomly weighted sums of WOD random variables and application to nonparametric regression models

This paper is devoted to the strong law for randomly weighted sums random variables. First, based on the properties of slowly varying functions and the de Bruijn conjugates, we prove a general result on the Marcinkiewicz-Zygmund strong law of large numbers for randomly weighted sums of widely orthant dependent (WOD, in short) random variables with general norming constants. In the second part of this paper, we establish almost sure consistency and convergence rate for the estimator of nonparametric regression model based on widely orthant dependent errors. The simulations to study the numerical performance of the consistency for the nearest neighbour weight function estimator are given to support the theoretical results.