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A Characterization of Probabilities with Full Support and the Laplace Method

  • Simone Cerreia-Vioglio
  • Fabio MaccheroniEmail author
  • Massimo Marinacci
Article
  • 65 Downloads

Abstract

We show that a probability measure on a metric space has full support, if, and only if, the set of all probability measures, that are absolutely continuous with respect to it, is dense in the set of all Borel probability measures. We illustrate the result through a general version of Laplace’s method, which in turn leads to general stochastic convergence to global maxima.

Keywords

Absolute continuity Support of a measure Laplace method 

Mathematics Subject Classification

46N30 46N10 60B05 

Notes

Acknowledgements

We thank Giacomo Cattelan, Ludovica Ciasullo, and Isabella Morgan Wolfskeil for excellent research assistance. Simone Cerreia-Vioglio, and Fabio Maccheroni and Massimo Marinacci gratefully acknowledge the financial support of ERC (Grants SDDM-TEA and INDIMACRO, respectively).

References

  1. 1.
    Romeijn, H.E., Smith, R.L.: Simulated annealing for constrained global optimization. J. Glob. Optim. 5, 101–126 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hiriart-Urruty, J.-B.: Conditions for global optimality. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 1–26. Springer, New York (1995)Google Scholar
  3. 3.
    De Bruin, B.: Explaining Games: The Epistemic Programme in Game Theory. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dekel, E., Siniscalchi, M.: Epistemic game theory. In: Young, P., Zamir, S. (eds.) Handbook of Game Theory, vol. 4, pp. 629–702. Elsevier, New York (2014)Google Scholar
  5. 5.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, New York (2006)zbMATHGoogle Scholar
  6. 6.
    Phelps, R.R.: Lectures on Choquet’s Theorem, 2nd edn. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Burzoni, M., Frittelli, M., Maggis, M.: Universal arbitrage aggregator in discrete-time markets under uncertainty. Finance Stoch. 20, 1–50 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Parpas, P., Rustem, B.: Laplace method and applications to optimization problems. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1818–1822. Springer, New York (2009)CrossRefGoogle Scholar
  9. 9.
    Hwang, C.R.: Laplace’s method revisited: weak convergence of probability measures. Ann. Probab. 8, 1177–1182 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (2011)zbMATHGoogle Scholar
  11. 11.
    Dal Maso, G.: An Introduction to \(\varGamma \)-Convergence. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  12. 12.
    Pincus, M.: A closed form solution of certain programming problems. Oper. Res. 16, 690–694 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pincus, M.: A Monte Carlo method for the approximate solution of certain types of constrained optimization problems. Oper. Res. 18, 1225–1228 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Simone Cerreia-Vioglio
    • 1
  • Fabio Maccheroni
    • 1
    Email author
  • Massimo Marinacci
    • 1
  1. 1.IGIERBocconi UniversityMilanItaly

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