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An Abstract Law of Large Numbers

Author

Listed:
  • Nabil I. Al-Najjar

    (Northwestern University)

  • Luciano Pomatto

    (California Institute of Technology)

Abstract

We study independent random variables (Zi)i∈I 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 ∫IZidν(i)${\int }_I Z_i d\nu (i)$. We establish that any ν that guarantees the measurability of ∫IZidν(i)${\int }_I Z_i d\nu (i)$ satisfies the following law of large numbers: for any collection (Zi)i∈I of uniformly bounded and independent random variables, almost surely the realized average ∫IZidν(i)${\int }_I Z_i d\nu (i)$ equals the average expectation ∫IE[Zi]dν(i)${\int }_I E[Z_i]d\nu (i)$.

Suggested Citation

  • Nabil I. Al-Najjar & Luciano Pomatto, 2020. "An Abstract Law of Large Numbers," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 1-12, February.
    • Handle: RePEc:spr:sankha:v:82:y:2020:i:1:d:10.1007_s13171-018-00162-z
    DOI: 10.1007/s13171-018-00162-z
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    References listed on IDEAS

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    1. Judd, Kenneth L., 1985. "The law of large numbers with a continuum of IID random variables," Journal of Economic Theory, Elsevier, vol. 35(1), pages 19-25, February.
    2. Gilboa, Itzhak & Matsui, Akihiko, 1992. "A model of random matching," Journal of Mathematical Economics, Elsevier, vol. 21(2), pages 185-197.
    3. Harald Uhlig, 1996. "A law of large numbers for large economies (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 41-50.
    4. Feldman, Mark & Gilles, Christian, 1985. "An expository note on individual risk without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 35(1), pages 26-32, February.
    5. Al-Najjar, Nabil I., 2008. "Large games and the law of large numbers," Games and Economic Behavior, Elsevier, vol. 64(1), pages 1-34, September.
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    Cited by:

    1. Al-Najjar, Nabil I. & Pomatto, Luciano, 2020. "Aggregate risk and the Pareto principle," Journal of Economic Theory, Elsevier, vol. 189(C).

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