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November 2008 Trivial intersection of σ-fields and Gibbs sampling
Patrizia Berti, Luca Pratelli, Pietro Rigo
Ann. Probab. 36(6): 2215-2234 (November 2008). DOI: 10.1214/07-AOP387

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 (Xn, Yn)n≥0 be the Gibbs chain for P. Then, the SLLN holds for (Xn, Yn) if and only if $\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or equivalently if and only if P(XU)P(YV)=0 whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and P(U×V)=P(Uc×Vc)=0. The latter condition is also equivalent to ergodicity of (Xn, Yn), 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.

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Patrizia Berti. Luca Pratelli. Pietro Rigo. "Trivial intersection of σ-fields and Gibbs sampling." Ann. Probab. 36 (6) 2215 - 2234, November 2008. https://doi.org/10.1214/07-AOP387

Information

Published: November 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1159.60007
MathSciNet: MR2478681
Digital Object Identifier: 10.1214/07-AOP387

Subjects:
Primary: 60A05, 60A10, 60J22, 65C05

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 6 • November 2008
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