Asymptotic properties of random weighted empirical distribution function

This paper studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X 1, X 2,..., 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.