234 Boolean Analysis — Inputs
234.1 Overview
The input moment conditions for fourth-order invariance are packaged together and shown to depend only on the input distribution.
234.2 Declarations
Let \(X\) be a real function on a measured space \((\Omega ,\mu )\). It satisfies the invariance moment conditions when \(\mu \) is a probability measure, \(X\) is measurable up to a null set with finite fourth absolute moment, \(\mathbb E X=\mathbb E X^3=0\), \(\mathbb E X^2=1\), and \(\mathbb E X^4\le 9\).
Let \(X\) and \(Y\) be real random variables on measured spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), where \(\nu \) is a probability measure. If \(X\) satisfies the invariance moment conditions and \(X\) and \(Y\) have the same distribution, then \(Y\) is measurable up to a null set with finite fourth absolute moment and moments \(\mathbb EY=\mathbb EY^3=0\), \(\mathbb EY^2=1\), and \(\mathbb EY^4\le 9\); thus \(Y\) satisfies the same moment conditions.