The Annals of Probability

Trivial intersection of σ-fields and Gibbs sampling

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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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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Ann. Probab., Volume 36, Number 6 (2008), 2215-2234.

First available in Project Euclid: 19 December 2008

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Zentralblatt MATH identifier

Primary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60J22: Computational methods in Markov chains [See also 65C40] 65C05: Monte Carlo methods

Ergodicity Gibbs sampler iterated conditional expectation Markov chain strong law of large numbers sufficiency


Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Trivial intersection of σ -fields and Gibbs sampling. Ann. Probab. 36 (2008), no. 6, 2215--2234. doi:10.1214/07-AOP387.

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