×

0-1 laws for regular conditional distributions. (English) Zbl 1118.60004

Summary: Let \((\Omega,{\mathcal B},P)\) be a probability space, \({\mathcal A}\subset {\mathcal B}\) a sub-\(\sigma\)-field, and \(\mu\) a regular conditional distribution for \(P\) given \({\mathcal A}\). Necessary and sufficient conditions for \(\mu(\omega)(A)\) to be 0-1, for all \(A\in{\mathcal A}\) and \(\omega\in A_0\), where \(A_0\in{\mathcal A}\) and \(P(A_0)=1\), are given. Such conditions apply, in particular, when \({\mathcal A}\) is a tail sub-\(\sigma\)-field. Let \(H(\omega)\) denote the \({\mathcal A}\)-atom including the point \(\omega\in\Omega\). Necessary and sufficient conditions for \(\mu (\omega)(H(\omega))\) to be 0-1, for all \(\omega\in A_0\), are also given. If \((\Omega,{\mathcal B})\) is a standard space, the latter 0-1 law is true for various classically interesting sub-\(\sigma\)-fields \({\mathcal A}\), including tail, symmetric, invariant, as well as some sub-\(\sigma\)-fields connected with continuous time processes.

MSC:

60A10 Probabilistic measure theory
60A05 Axioms; other general questions in probability
60F20 Zero-one laws
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Berti, P. and Rigo, P. (1999). Sufficient conditions for the existence of disintegrations. J. Theoret. Probab. 12 75–86. · Zbl 0926.60012 · doi:10.1023/A:1021792409934
[2] Blackwell, D. (1956). On a class of probability spaces. In Proc. Third Berkeley Symp. Math. Statist. Probab. 1–6. Univ. California Press, Berkeley. · Zbl 0073.12301
[3] Blackwell, D. and Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. Ann. Math. Statist. 34 223–225. · Zbl 0122.13202 · doi:10.1214/aoms/1177704259
[4] Blackwell, D. and Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. Ann. Probab. 3 741–752. · Zbl 0348.60003 · doi:10.1214/aop/1176996261
[5] Diaconis, P. and Freedman, D. (1984). Partial exchangeability and sufficiency. In Proc. of the Indian Statistical Institute Golden Jubilee International Conference on Statistics : Applications and New Directions 205–236. Indian Statist. Inst., Calcutta.
[6] Dubins, L. E. and Freedman, D. (1964). Measurable sets of measures. Pacific J. Math. 14 1211–1222. · Zbl 0148.03002 · doi:10.2140/pjm.1964.14.1211
[7] Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probab. 6 705–730. · Zbl 0403.62009 · doi:10.1214/aop/1176995424
[8] Fremlin, D. H. (1981). Measurable functions and almost continuous functions. Manuscripta Math. 33 387–405. · Zbl 0459.28010 · doi:10.1007/BF01798235
[9] Maitra, A. (1977). Integral representations of invariant measures. Trans. Amer. Math. Soc. 229 209–225. · Zbl 0357.28020 · doi:10.2307/1998506
[10] Sazonov, V. V. (1965). On perfect measures. Amer. Math. Soc. Transl. Ser. (2) 48 229–254. · Zbl 0152.04301
[11] Seidenfeld, T., Schervish, M. J. and Kadane, J. (2001). Improper regular conditional distributions. Ann. Probab. 29 1612–1624. · Zbl 1017.60007 · doi:10.1214/aop/1015345764
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.