Trivial intersection of σ-fields and Gibbs sampling

Abstract

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying P(F)∈{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-σ-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≥0 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∈U)P(Y∈V)=0 whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and P(U×V)=P(U^c×V^c)=0. The latter condition is also equivalent to ergodicity of (X_n, Y_n), on a certain subset S₀⊂Ω, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and P absolutely continuous with respect to a product measure.