Advertisement

Sankhya A

pp 1–12 | Cite as

An Abstract Law of Large Numbers

  • Nabil I. Al-Najjar
  • Luciano PomattoEmail author
Article
  • 9 Downloads

Abstract

We study independent random variables (Zi)iI aggregated by integrating with respect to a nonatomic and finitely additive probability ν over the index set I. We analyze the behavior of the resulting random average \({\int }_I Z_i d\nu (i)\). We establish that any ν that guarantees the measurability of \({\int }_I Z_i d\nu (i)\) satisfies the following law of large numbers: for any collection (Zi)iI of uniformly bounded and independent random variables, almost surely the realized average \({\int }_I Z_i d\nu (i)\) equals the average expectation \({\int }_I E[Z_i]d\nu (i)\).

Keywords

Finitely additive probabilities Measure theory Measurability 

AMS (2000) subject classification

Primary 28A25 Secondary 60F15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. Al-Najjar, N.I. (2008). Large games and the law of large numbers. Games Econom. Behav. 64, 1, 1–34.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Berti, P. and Rigo, P. (2006). Finitely additive uniform limit theorems. Sankhyā: The Indian Journal of Statistics (2003–2007) 68, 1, 24–44.MathSciNetzbMATHGoogle Scholar
  3. Christensen, J.P.R. (1971). Borel structures and a topological zero-one law. Math. Scand. 29, 2, 245–255.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Christensen, J.P.R. (1974). Topology and Borel structure. North Holland, Amsterdam.Google Scholar
  5. Dalal, S. (1978). A note on the adequacy of mixtures of Dirichlet processes. Sankhyā: The Indian Journal of Statistics, Series A (1961–2002) 40, 2, 185–191.MathSciNetzbMATHGoogle Scholar
  6. Feldman, M. and Gilles, C. (1985). An expository note on individual risk without aggregate uncertainty. J. Econ. Theory 35, 1, 26–32.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fisher, A. (1987). Convex-invariant means and a pathwise central limit theorem. Adv. Math. 63, 3, 213–246.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fremlin, D.H. and Talagrand, M. (1979). A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means. Mathematische Zeitschrift 168, 2, 117–142.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gangopadhyay, S. and Rao, B.V. (1999). On the hewitt-savage zero one law in the strategic setup. Sankhyā: The Indian Journal of Statistics, Series A, 153–165.Google Scholar
  10. Gilboa, I. and Matsui, A. (1992). A model of random matching. J. Math. Econ. 21, 2, 185–197.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Judd, K.L. (1985). The law of large numbers with a continuum of iid random variables. J. Econ. Theory 35, 1, 19–25.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kadane, J.B. and O’Hagan, A. (1995). Using finitely additive probability: uniform distributions on the natural numbers. J. Am. Stat. Assoc. 90, 430, 626–631.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kallianpur, G. and Karandikar, R.L. (1988). White Noise Theory of Prediction, Filtering and Smoothing, 3. CRC Press, Boca Raton.zbMATHGoogle Scholar
  14. Kamae, T., Krengel, U. and O’Brien, G.L. (1977). Stochastic inequalities on partially ordered spaces. The Annals of Probability 5, 6, 899–912.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Karandikar, R.L. (1982). A general principle for limit theorems in finitely additive probability. Trans. Am. Math. Soc. 273, 2, 541–550.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Larson, P.B. (2009). The filter dichotomy and medial limits. J. Math. Log. 9, 02, 159–165.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Maharam, D. (1976). Finitely additive measures on the integers. Sankhyā: The Indian Journal of Statistics, Series A (1961–2002) 38, 1, 44–59.MathSciNetzbMATHGoogle Scholar
  18. Meyer, P.-A. (1973). Limites médiales, d’après Mokobodzki, Séminaire de probabilités VII. Springer, Berlin, p. 198–204.Google Scholar
  19. Nutz, M. et al. (2012). Pathwise construction of stochastic integrals. Electron. Commun. Probab., 17.Google Scholar
  20. Paul, E. (1962). Density in the light of probability theory. Sankhyā: The Indian Journal of Statistics, Series A (1961–2002) 24, 2, 103–114.MathSciNetzbMATHGoogle Scholar
  21. Purves, R. and Sudderth, W. (1983). Finitely additive zero-one laws. Sankhyā: The Indian Journal of Statistics, Series A (1961–2002) 45, 1, 32–37.MathSciNetzbMATHGoogle Scholar
  22. Uhlig, H. (1996). A law of large numbers for large economies. Econ. Theory 8, 1, 41–50.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Department of Managerial Economics and Decision Sciences, Kellogg School of ManagementNorthwestern UniversityEvanstonUSA
  2. 2.Division of the Humanities and Social SciencesCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations