Trivial intersection of $σ$-fields and Gibbs sampling

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro

Trivial intersection of $σ$-fields and Gibbs sampling

Authors

Berti, Patrizia

Pratelli, Luca

Rigo, Pietro

Description

Let $(Ω,\mathcal{F},P)$ 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 $\overline{\mathcal{G}}=σ(\mathcal{G}\cup\mathcal {N})$. 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 $(Ω,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ 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 $\overline{σ(X)}\cap\overline{σ(Y)}=\mathcal{N}$, 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 $S_0\subsetΩ$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure.
Published in at http://dx.doi.org/10.1214/07-AOP387 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)