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The Markov moment problem and de Finetti’s theorem: Part II

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Abstract.

This paper gives an abstract version of de Finetti’s theorem that characterizes mixing measures with L p densities. The general setting is reviewed; after the theorem is proved, it is specialized to coin tossing and to exponential random variables. Laplace transforms of bounded densities are characterized, and inversion formulas are discussed.

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References

  1. Aldous, D.J.: Exchangeability and related topics. École d’été de probabilités de Saint-Flour XIII. edité par P. L. Hennequin. Lecture notes in mathematics, 1117, Springer-Verlag, 1985

  2. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic analysis on semigroups: theory of positive definite and related functions. Springer Graduate Texts in Mathematics, 100, 1984

  3. Diaconis, P., Freedman, D.: Partial exchangeability and sufficiency. In: J. K. Ghosh, J. Roy, (eds.), Proc. Indian Statist. Assoc. Golden Jubilee: Applications And New Directions. Indian Statist. Assoc., Calcutta, 1984, pp. 205–236

  4. Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II, Second edition, Wiley, New York, 1971

  5. Freedman, D.A.: Invariants under mixing which generalize de Finetti’s theorem. Ann. Math. Stat. 34, 1194–1216 (1963)

    MATH  Google Scholar 

  6. Georgii, H.O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin, 1988

  7. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York, 1969

  8. Hirschman, I.I., Widder, D.V.: The Convolution Transform. Princeton University Press, 1955

  9. Hunt, G.: Markoff chains and Martin boundaries. Ill. J. Math. 4, 313–40 (1960)

    MATH  Google Scholar 

  10. Kallenberg, O.: Multivariate sampling and the estimation problem for exchangeable arrays. J. Theor. Prob. 12, 859–83 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lanford, O., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13, 194–215 (1969)

    Google Scholar 

  12. Lauritzen, S.L.: Extremal Families and Systems of Sufficient Statistics. Springer Lecture Notes in Statistics, 49, 1988

  13. Lehmann, E.L.: Testing Statistical Hypotheses. 2nd ed. Wadsworth & Brooks/Cole, 1991

  14. Martin-Löf, P.: Repetitive structures and the relation between canonical and micro-canonical distributions in statistics and statistical mechanics. In: O. Barndorff-Nielsen, P. Blaesild, G. Schou, (eds.), Proceedings of Conference on Foundational Questions in Statistical Inference. Aarhus, 1974

  15. Oxtoby, J.C.: Ergodic sets. Bull. Amer. Math. Soc. 58, 116–36 (1952)

    MathSciNet  MATH  Google Scholar 

  16. Ressel, P.: De Finetti-type theorems: An analytical approach. Ann. Prob. 13, 898–922 (1985)

    MathSciNet  MATH  Google Scholar 

  17. Ruelle, D.: Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics. Cambridge University Press, 1984

  18. Schervish, M.J.: Theory of Statistics. Springer, 1995

  19. Schoenberg, I.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–41 (1938a)

  20. Schoenberg, I.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–36 (1938b)

    Google Scholar 

  21. Widder, D. V.: The Laplace Transform. First printing, 1941, Princeton University Press, 1946

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Authors and Affiliations

  1. Department of Mathematics and of Statistics, Stanford University, Stanford, CA 94305, USA

    Persi Diaconis

  2. Department of Statistics and of Mathematics, University of California, Berkeley, CA 94720-3860, USA

    David Freedman

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  1. Persi Diaconis
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  2. David Freedman
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Correspondence to David Freedman.

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Diaconis, P., Freedman, D. The Markov moment problem and de Finetti’s theorem: Part II. Math. Z. 247, 201–212 (2004). https://doi.org/10.1007/s00209-003-0636-6

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  • DOI: https://doi.org/10.1007/s00209-003-0636-6

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