The Annals of Probability

Convergence in distribution of nonmeasurable random elements

Patrizia Berti and Pietro Rigo

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Abstract

A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 365-379.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415839

Digital Object Identifier
doi:10.1214/aop/1078415839

Mathematical Reviews number (MathSciNet)
MR2040786

Zentralblatt MATH identifier
1049.60004

Subjects
Primary: 60B10: Convergence of probability measures 60A05: Axioms; other general questions

Keywords
Convergence in distribution empirical process exchangeability extension finitely additive probability measure measurability

Citation

Berti, Patrizia; Rigo, Pietro. Convergence in distribution of nonmeasurable random elements. Ann. Probab. 32 (2004), no. 1A, 365--379. doi:10.1214/aop/1078415839. https://projecteuclid.org/euclid.aop/1078415839


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