Asymptotic Properties of Random Weighted Empirical Distribution Function

Abstract

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ⋅⋅⋅, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.

Keywords:

• Central limit theorem
• Empirical distribution function
• Glivenko–Cantelli theorem
• Random weighting method

Mathematics Subject Classification:

• 60E05; 60F05