On characterization of a.s. convergence of all conditionings for sequences of random variables

On characterization of a.s. convergence of all conditionings for sequences of random variables

We prove that in a non-atomic probability space, for a sequence of any integrable r.v. ( ) we have the following equivalence: E( |U) → 0 a.s. for any σ-field U of events iff → 0 a.s. and E sup | | < ∞.

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Veröffentlicht in:Demonstratio mathematica 2012-06, Vol.45 (2), p.455-462
1. Verfasser: Paszkiewicz, Adam
Format: Artikel
Sprache:eng
Schlagworte:
28A20
40A30
almost everywhere convergences
conditional expectations
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Zusammenfassung:We prove that in a non-atomic probability space, for a sequence of any integrable r.v. ( ) we have the following equivalence: E( |U) → 0 a.s. for any σ-field U of events iff → 0 a.s. and E sup | | < ∞.
ISSN:2391-4661
2391-4661
DOI:10.1515/dema-2013-0376