Abstract
Let \({(\Omega, \mathcal{F}, P)}\) be a probability space. For each \({\mathcal{G}\subset\mathcal{F}}\), define \({\overline{\mathcal{G}}}\) as the σ-field generated by \({\mathcal{G}}\) and those sets \({F\in \mathcal{F}}\) satisfying \({P(F)\in\{0,1\}}\). Conditions for P to be atomic on \({\cap_{i=1}^k\overline{\mathcal{A}_i}}\), with \({\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}\) sub-σ-fields, are given. Conditions for P to be 0-1-valued on \({\cap_{i=1}^k \overline{\mathcal{A}_i}}\) are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.
References
Berti P., Pratelli L., Rigo P.: Asymptotic behaviour of the empirical process for exchangeable data. Stoch. Proc. Appl. 116, 337–344 (2006)
Berti P., Pratelli L., Rigo P.: Skorohod representation on a given probability space. Prob. Theo. Rel. Fields 137, 277–288 (2007)
Berti P., Pratelli L., Rigo P.: Trivial intersection of σ-fields and Gibbs sampling. Ann. Probab. 36, 2215–2234 (2008)
Bhaskara Rao K.P.S., Bhaskara Rao M.: Theory of Charges. Academic Press, London (1983)
Burkholder D.L., Chow Y.S.: Iterates of conditional expectation operators. Proc. Am. Math. Soc. 12, 490–495 (1961)
Delyon B., Delyon F.: Generalization of von Neumann’s spectral sets and integral representation of operators. Bull. Soc. Math. Fr. 127, 25–41 (1999)
Diaconis, P., Khare, K., Saloff-Coste, L.: Stochastic alternating projections. Dept. of Statistics, Stanford University, currently available at: http://www-stat.stanford.edu/~cgates/PERSI/papers/altproject-2.pdf (2007, preprint)
Halmos P.R.: On the set of values of a finite measure. Bull. Am. Math. Soc. 53, 138–141 (1947)
Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, vol. 2. Springer, Heidelberg (1991)
San Martin E., Mouchart M., Rolin J.M.: Ignorable common information, null sets and Basu’s first theorem. Sankhya 67, 674–698 (2005)
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Berti, P., Pratelli, L. & Rigo, P. Atomic intersection of σ-fields and some of its consequences. Probab. Theory Relat. Fields 148, 269–283 (2010). https://doi.org/10.1007/s00440-009-0230-x
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DOI: https://doi.org/10.1007/s00440-009-0230-x
Keywords
- Atomic probability measure
- Gibbs sampling
- Graphical models
- Intersection property
- Iterated conditional expectations
Mathematics Subject Classification (2000)
- 60A10
- 60J22
- 62B99