Statistical Methods & Applications

, Volume 23, Issue 4, pp 483–500 | Cite as

A unifying view on some problems in probability and statistics

  • Patrizia Berti
  • Luca Pratelli
  • Pietro Rigo


Let \(L\) be a linear space of real random variables on the measurable space \((\varOmega ,\mathcal {A})\). Conditions for the existence of a probability \(P\) on \(\mathcal {A}\) such that \(E_P|X|<\infty \) and \(E_P(X)=0\) for all \(X\in L\) are provided. Such a \(P\) may be finitely additive or \(\sigma \)-additive, depending on the problem at hand, and may also be requested to satisfy \(P\sim P_0\) or \(P\ll P_0\) where \(P_0\) is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability \(P\) as above. Among them, we mention de Finetti’s coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions.


Compatibility of conditional distributions De Finetti’s coherence principle Equivalent martingale measure Equivalent probability measure with given marginals Finitely additive probability Fundamental theorem of asset pricing 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed Applicata “G. Vitali”Universita’ di Modena e Reggio-EmiliaModenaItaly
  2. 2.Accademia NavaleLivornoItaly
  3. 3.Dipartimento di Matematica “F. Casorati”Universita’ di PaviaPaviaItaly

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