An Abstract Law of Large Numbers

Nabil Al-Najjar, Luciano Pomatto*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

We study independent random variables (Z i ) 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 ∫ I Z i dν(i). We establish that any ν that guarantees the measurability of ∫ I Z i dν(i) satisfies the following law of large numbers: for any collection (Z i ) i∈I of uniformly bounded and independent random variables, almost surely the realized average ∫ I Z i dν(i) equals the average expectation ∫ I E[Z i ] dν(i).

Original languageEnglish (US)
JournalSankhya A
DOIs
StateAccepted/In press - Jan 1 2019

Fingerprint

Law of large numbers
Independent Random Variables
Measurability
Random variables
Guarantee

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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An Abstract Law of Large Numbers. / Al-Najjar, Nabil; Pomatto, Luciano.

In: Sankhya A, 01.01.2019.

Research output: Contribution to journalArticle

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AB - We study independent random variables (Z i ) 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 ∫ I Z i dν(i). We establish that any ν that guarantees the measurability of ∫ I Z i dν(i) satisfies the following law of large numbers: for any collection (Z i ) i∈I of uniformly bounded and independent random variables, almost surely the realized average ∫ I Z i dν(i) equals the average expectation ∫ I E[Z i ] dν(i).

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