Trivial intersection of \(\sigma \)-fields and Gibbs sampling. (English) Zbl 1159.60007

In this paper the attention focuses on necessary and sufficient conditions for \(\bar\mathcal A\bigcap\bar\mathcal B=\overline{\mathcal A\bigcap\mathcal B}\), where \(\mathcal A, \mathcal B\subset\mathcal F\) are sub \(\sigma\)-fields and \((\Omega,\mathcal F,\mathcal P)\) is a probability space. 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 \(\overline{\sigma(X)}\bigcap\overline{\sigma(Y)}=\mathcal N\), 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.


60A05 Axioms; other general questions in probability
60A10 Probabilistic measure theory
60J22 Computational methods in Markov chains
65C05 Monte Carlo methods
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