Convergence Results for Approximation with independent Variables

Abstract

For a square integrable $m$-dimensional random variable $X$ on a probability space $(Ω,\Fc,\Pr)$ and a sub sigma algebra $\Ac$, we show that there is a constructive way to represent $X-\Er[X\mid\Ac]$ as the sum of a series of variables that are independent of $\Ac$.