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A unifying view on some problems in probability and statistics

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  • Patrizia Berti
  • Luca Pratelli
  • Pietro Rigo

Abstract

Let $$L$$ L be a linear space of real random variables on the measurable space $$(\varOmega ,\mathcal {A})$$ ( Ω , A ) . Conditions for the existence of a probability $$P$$ P on $$\mathcal {A}$$ A such that $$E_P|X|>\infty $$ E P | X | > ∞ and $$E_P(X)=0$$ E P ( X ) = 0 for all $$X\in L$$ X ∈ L are provided. Such a $$P$$ P may be finitely additive or $$\sigma $$ σ -additive, depending on the problem at hand, and may also be requested to satisfy $$P\sim P_0$$ P ∼ P 0 or $$P\ll P_0$$ P ≪ P 0 where $$P_0$$ P 0 is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability $$P$$ 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. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Patrizia Berti & Luca Pratelli & Pietro Rigo, 2014. "A unifying view on some problems in probability and statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(4), pages 483-500, November.
    • Handle: RePEc:spr:stmapp:v:23:y:2014:i:4:p:483-500
    DOI: 10.1007/s10260-014-0272-9
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    References listed on IDEAS

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    1. Berti, Patrizia & Dreassi, Emanuela & Rigo, Pietro, 2014. "Compatibility results for conditional distributions," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 190-203.
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    Cited by:

    1. Michael Nielsen, 2019. "On linear aggregation of infinitely many finitely additive probability measures," Theory and Decision, Springer, vol. 86(3), pages 421-436, May.
    2. Berti Patrizia & Pratelli Luca & Rigo Pietro & Spizzichino Fabio, 2015. "Equivalent or absolutely continuous probability measures with given marginals," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-12, May.

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