Convergence Results for Approximation with independent Variables

TL;DR

This paper presents a constructive method to represent the difference between a square integrable random variable and its conditional expectation as a series of independent variables, enhancing understanding of approximation techniques in probability theory.

Contribution

It introduces a novel constructive approach to decompose a random variable into a series of independent components relative to a sub sigma algebra.

Findings

Representation of $X - ext{E}[X| ext{A}]$ as a series of independent variables.

Provides a constructive method for approximation in probability spaces.

Enhances theoretical understanding of independence and conditional expectation.

Abstract

For a square integrable $m$-dimensional random variable $X$ on a probability space $(\Omega,\Fc,\Pr)$ 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 $\Ac$.