## 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.

### MSC:

 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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### References:

 [1] Berti, P., Pratelli, L. and Rigo, P. (2007). Skorohod representation on a given probability space. Probab. Theory Related Fields 137 277-288. · Zbl 1106.60003 [2] Burkholder, D. L. and Chow, Y. S. (1961). Iterates of conditional expectation operators. Proc. Amer. Math. Soc. 12 490-495. JSTOR: · Zbl 0106.33201 [3] Burkholder, D. L. (1961). Sufficiency in the undominated case. Ann. Math. Statist. 32 1191-1200. · Zbl 0221.62003 [4] Burkholder, D. L. (1962). Successive conditional expectations of an integrable function. Ann. Math. Statist. 33 887-893. · Zbl 0128.12602 [5] Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci. 23 151-178. · Zbl 1327.62058 [6] Diaconis, P., Freedman, D., Khare, K. and Saloff-Coste, L. (2007). Stochastic alternating projections. Preprint, Dept. Statistics, Stanford Univ. Currently available at http://www-stat.stanford.edu/ cgates/PERSI/papers/altproject-2.pdf. · Zbl 1268.60098 [7] Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312-334. · Zbl 1127.60309 [8] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics . Springer, New York. · Zbl 0991.65001 [9] Meyn, S. P. and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability . Springer, New York. · Zbl 0925.60001 [10] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701-1762. With discussion and a rejoinder by the author. · Zbl 0829.62080
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