An Abstract Law of Large Numbers


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 \({\int }_I E[Z_i]d\nu (i)\).

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  1. Al-Najjar, N.I. (2008). Large games and the law of large numbers. Games Econom. Behav.64, 1, 1–34.

    MathSciNet  Article  Google 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.

    MathSciNet  MATH  Google Scholar 

  3. Christensen, J.P.R. (1971). Borel structures and a topological zero-one law. Math. Scand.29, 2, 245–255.

    MathSciNet  Article  Google Scholar 

  4. Christensen, J.P.R. (1974). Topology and Borel structure. North Holland, Amsterdam.

  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.

    MathSciNet  MATH  Google Scholar 

  6. Feldman, M. and Gilles, C. (1985). An expository note on individual risk without aggregate uncertainty. J. Econ. Theory35, 1, 26–32.

    MathSciNet  Article  Google Scholar 

  7. Fisher, A. (1987). Convex-invariant means and a pathwise central limit theorem. Adv. Math.63, 3, 213–246.

    MathSciNet  Article  Google 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 Zeitschrift168, 2, 117–142.

    MathSciNet  Article  Google 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.

  10. Gilboa, I. and Matsui, A. (1992). A model of random matching. J. Math. Econ.21, 2, 185–197.

    MathSciNet  Article  Google Scholar 

  11. Judd, K.L. (1985). The law of large numbers with a continuum of iid random variables. J. Econ. Theory35, 1, 19–25.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google Scholar 

  13. Kallianpur, G. and Karandikar, R.L. (1988). White Noise Theory of Prediction, Filtering and Smoothing, 3. CRC Press, Boca Raton.

    Google Scholar 

  14. Kamae, T., Krengel, U. and O’Brien, G.L. (1977). Stochastic inequalities on partially ordered spaces. The Annals of Probability5, 6, 899–912.

    MathSciNet  Article  Google Scholar 

  15. Karandikar, R.L. (1982). A general principle for limit theorems in finitely additive probability. Trans. Am. Math. Soc.273, 2, 541–550.

    MathSciNet  Article  Google Scholar 

  16. Larson, P.B. (2009). The filter dichotomy and medial limits. J. Math. Log.9, 02, 159–165.

    MathSciNet  Article  Google 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.

    MathSciNet  MATH  Google 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.

  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.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  22. Uhlig, H. (1996). A law of large numbers for large economies. Econ. Theory8, 1, 41–50.

    MathSciNet  Article  Google Scholar 

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Correspondence to Luciano Pomatto.

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Al-Najjar, N.I., Pomatto, L. An Abstract Law of Large Numbers. Sankhya A 82, 1–12 (2020).

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  • Finitely additive probabilities
  • Measure theory
  • Measurability

AMS (2000) subject classification

  • Primary 28A25
  • Secondary 60F15