An abstract law of large numbers

Nabil I. Al-Najjar, Luciano Pomatto*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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) We establish that any ν that guarantees the measurability of ∫IZidν(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) equals the average expectation ∫IE[Zi]dν(i).

Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalSankhya: The Indian Journal of Statistics
Volume82A
StatePublished - 2020

Keywords

  • Finitely additive probabilities
  • Measurability
  • Measure theory

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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