An Abstract Law of Large Numbers

Nabil I. Al-Najjar, Luciano Pomatto*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

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)
JournalSankhya A
Volume82
Issue number1
DOIs
StatePublished - Feb 1 2020

Keywords

  • Finitely additive probabilities
  • Measurability
  • Measure theory
  • Primary 28A25
  • Secondary 60F15

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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