239 Boolean Analysis — Invariance — Sequence
239.1 Overview
A sequence of hybrid input families gives the summed replacement bound. Equality of independent coordinate laws also determines the joint vector law.
239.2 Declarations
For integers \(n,r\ge 0\) and an index \(i\in \{ 0,\ldots ,n-1\} \), the hybrid selection chooses the right copy of \(i\) if \(i\lt r\) and the left copy otherwise, as an element of two disjoint copies of the coordinate set.
Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two indexed families, \(0\le i\lt n\), and let \(r\ge 0\) be an integer. The hybrid family \(H^{(r)}\) has \(H^{(r)}_i=Z_{R,i}\) for \(i\lt r\) and \(H^{(r)}_i=Z_{L,i}\) for \(i\ge r\).
For integers \(n,r\ge 0\), the map sending \(i\in \{ 0,\ldots ,n-1\} \) to its right copy when \(i\lt r\) and its left copy otherwise is injective.
Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le i\lt n\), and define \(H^{(r)}_i=Z_{R,i}\) if \(i\lt r\) and \(Z_{L,i}\) otherwise. Then \(H^{(0)}_i=Z_{L,i}\) for every \(i\).
Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le i\lt n\), and define \(H^{(r)}_i=Z_{R,i}\) if \(i\lt r\) and \(Z_{L,i}\) otherwise. Then \(H^{(n)}_i=Z_{R,i}\) for every \(i\), including the empty family when \(n=0\).
Let \(Z_{L,j},Z_{R,j}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le j\lt n\), and put \(H^{(r)}_j=Z_{R,j}\) if \(j\lt r\) and \(Z_{L,j}\) otherwise. If \(i,j\in \{ 0,\ldots ,n-1\} \) and \(j\ne i\), then \(H^{(i)}_j=H^{(i+1)}_j\) as functions on \(\Omega \).
Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \((Z_{L,i},Z_{R,i})_{i=1}^n\) be a mutually independent family of \(2n\) real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\) and \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, then \(|\mathbb E\psi (F(Z_L))-\mathbb E\psi (F(Z_R))|\le (C/12)9^k\sum _i I_i(F)^2\).
Let \(I\) be a finite index set and let \((X_i)_{i\in I}\) and \((Y_i)_{i\in I}\) be real input families on probability spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), respectively. If each family is independent and \(X_i\) and \(Y_i\) are measurable up to null sets with the same distribution for every \(i\), then the random vectors \((X_i)_{i\in I}\) and \((Y_i)_{i\in I}\) have the same distribution.