Finitely additive mixtures of probability measures

Abstract Let D be a linear space of real bounded functions and P : D → R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P ( f ) = ∫ Q Q ( f ) Π ( d Q ) for each f ∈ D . This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

[1]  Equivalent or absolutely continuous probability measures with given marginals , 2015 .

[2]  Patrizia Berti,et al.  On Coherent Conditional Probabilities and Disintegrations , 2002, Annals of Mathematics and Artificial Intelligence.

[3]  Karel Prikry,et al.  On the existence of disintegrations , 1995 .

[4]  L. Dubins.  Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations , 1975 .

[5]  B. Finetti.  Sul significato soggettivo della probabilità , 1931 .

[6]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[7]  Walter Schachermayer,et al.  The Mathematics of Arbitrage , 2006 .

[8]  L. J. Savage,et al.  Symmetric measures on Cartesian products , 1955 .

[9]  Patrizia Berti,et al.  Convergence in distribution of nonmeasurable random elements , 2004 .

[10]  G. Gregory,et al.  Probability, Induction and Statistics , 1974 .

[11]  Gerhard Winkler,et al.  On the integral representation in convex noncompact sets of tight measures , 1978 .

[12]  L. Pratelli,et al.  Two versions of the fundamental theorem of asset pricing , 2014, 1402.3570.

[13]  Patrizia Berti,et al.  Weak disintegrability as a form of preservation of coherence , 1992 .

[14]  William D. Sudderth,et al.  Some Finitely Additive Probability , 1976 .

[15]  Eugenio Regazzini,et al.  De Finetti's Coherence and Statistical Inference , 1987 .

[16]  Bruce M. Hill,et al.  Parametric Models for AN: Splitting Processes and Mixtures , 1993 .

[17]  Erik Ellentuck,et al.  Finitely additive measures and the first digit problem , 1969 .

[18]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[19]  A. Maitra,et al.  Integral representations of invariant measures , 1977 .

[20]  Patrizia Berti,et al.  Sufficient Conditions for the Existence of Disintegrations , 1999 .

[21]  Luca Pratelli,et al.  A unifying view on some problems in probability and statistics , 2014, Stat. Methods Appl..

[22]  Gopinath Kallianpur,et al.  White Noise Theory of Prediction, Filtering and Smoothing , 1988 .

[23]  0–1 laws for regular conditional distributions , 2006, math/0606604.

[24]  Donato Michele Cifarelli,et al.  De Finetti''''s contributions to probability and statistics , 1996 .

[25]  David Lindley,et al.  How to Gamble if You Must. (Inequalities for Stochastic Processes) By Lester E. Dubins and Leonard J. Savage. Pp. xiv, 249. 102s. 1965. (McGraw-Hill Book Co.) , 1966, The Mathematical Gazette.

[26]  P. M. Williams,et al.  Notes on conditional previsions , 2007, Int. J. Approx. Reason..

[27]  D. Maharam.  Consistent Extensions of Linear Functionals and of Probability Measures , 1972 .

[28]  David A. Freedman,et al.  Exchangeable processes need not be mixtures of independent, identically distributed random variables , 1979 .

[29]  The marginal problem in arbitrary product spaces , 1996 .

[30]  V. Varadarajan,et al.  Groups of automorphisms of Borel spaces , 1963 .

[31]  William D. Sudderth,et al.  On Finitely Additive Priors, Coherence, and Extended Admissibility , 1978 .

[32]  R. H. Farrell.  Representation of invariant measures , 1962 .

[33]  Patrizia Berti,et al.  Gluing lemmas and Skorohod representations , 2015 .