Trivial intersection of $\sigma$-fields and Gibbs sampling

TL;DR

This paper characterizes when the intersection of certain sigma-fields is trivial and applies these results to establish conditions for the strong law of large numbers and ergodicity in Gibbs sampling.

Contribution

It provides necessary and sufficient conditions for trivial intersections of sigma-fields and applies these to analyze convergence properties of Gibbs samplers.

Findings

Conditions for trivial intersection of sigma-fields are established.

The strong law of large numbers holds under specific independence conditions.

Ergodicity of Gibbs chains is characterized by these sigma-field conditions.

Abstract

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying $P(F)\in\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-$\sigma$-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose $X$ and $Y$ are the coordinate projections on $(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be the Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if $\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$. The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset $S_0\subset\Omega$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure.