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Atomic intersection of σ-fields and some of its consequences
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  • Published: 10 June 2009

Atomic intersection of σ-fields and some of its consequences

  • Patrizia Berti1,
  • Luca Pratelli2 &
  • Pietro Rigo3 

Probability Theory and Related Fields volume 148, pages 269–283 (2010)Cite this article

  • 89 Accesses

  • 3 Citations

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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.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata “G. Vitali”, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100, Modena, Italy

    Patrizia Berti

  2. Accademia Navale, viale Italia 72, 57100, Livorno, Italy

    Luca Pratelli

  3. Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100, Pavia, Italy

    Pietro Rigo

Authors
  1. Patrizia Berti
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  2. Luca Pratelli
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  3. Pietro Rigo
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Correspondence to Pietro Rigo.

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Cite this article

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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  • Received: 01 December 2007

  • Revised: 05 May 2009

  • Published: 10 June 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0230-x

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Keywords

  • Atomic probability measure
  • Gibbs sampling
  • Graphical models
  • Intersection property
  • Iterated conditional expectations

Mathematics Subject Classification (2000)

  • 60A10
  • 60J22
  • 62B99
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