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An Abstract Law of Large Numbers

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

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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). https://doi.org/10.1007/s13171-018-00162-z

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Keywords

  • Finitely additive probabilities
  • Measure theory
  • Measurability

AMS (2000) subject classification

  • Primary 28A25
  • Secondary 60F15