4 The 100% version of PFR
If \(X\) is a \(G\)-valued random variable, then the symmetry group \(\mathrm{Sym}[X]\) is the set of all \(h \in G\) such that \(X +h\) has the same distribution as \(X\).
If \(X\) is a \(G\)-valued random variable, then \(\mathrm{Sym}[X]\) is a subgroup of \(G\).
Direct verification of the group axioms.
If \(X\) is a \(G\)-valued random variable such that \(d[X ;X]=0\), and \(x,y \in G\) are such that \(P[X=x], P[X=y]{\gt}0\), then \(x-y \in \mathrm{Sym}[X]\).
Let \(X_1,X_2\) be independent copies of \(X\) (from Lemma 3.7). Let \(A\) denote the range of \(X\). From Lemma 3.11 and Lemma 3.10 we have
Observe from Lemma 2.12 that
and hence by Lemma 2.16
By Lemma 2.23, \(X_1-X_2\) and \(X_1\) are therefore independent, thus the law of \((X_1-X_2|X_1=x)\) does not depend on \(x \in A\). The claim follows.
If \(X\) is a \(G\)-valued random variable with \(d[X ;X]=0\), and \(x_0\) is a point with \(P[X=x_0] {\gt} 0\), then \(X-x_0\) is uniformly distributed on \(\mathrm{Sym}[X]\).
The law of \(X-x_0\) is invariant under \(\mathrm{Sym}[X]\), non-zero at the origin, and supported on \(\mathrm{Sym}[X]\), giving the claim.
Suppose that \(X\) is a \(G\)-valued random variable such that \(d[X ;X]=0\). Then there exists a subgroup \(H \leq G\) such that \(d[X ;U_H] = 0\).
Suppose that \(X_1,X_2\) are \(G\)-valued random variables such that \(d[X_1;X_2]=0\). Then there exists a subgroup \(H \leq G\) such that \(d[X_1;U_H] = d[X_2;U_H] = 0\).
Using Lemma 3.18 and Lemma 3.15 we have \(d[X_1;X_1]=0\), hence by Lemma 4.5 \(d[X_1;U_H]=0\) for some subgroup \(H\). By Lemma 3.18 and Lemma 3.15 again we also have \(d[X_2;U_H]\) as required.