A Weighted Central Limit Theorem for a Function-Indexed Sum with Random Point Masses

Abstract

Let F is a P-Donsker class of functions for a probability P on a sample space X. The purpose of the present note is to establish sufficient conditions for a functional central limit theorem for weighted sums of the form

$$\sum\limits_{{j = 1}}^{n} {{{\xi }_{{nj}}}{{\delta }_{{{{X}_{j}}}}}}$$

(1)

where ξ = (ξn nj,j = ₁, …,n, n = 1, 2, …) is a triangular array of row-independent random variables, and X₁, …, X _n are sampled iid and independent of ξ from P.