# An Abstract Law of Large Numbers

Nabil I. Al-Najjar and Luciano Pomatto ()
Nabil I. Al-Najjar: Northwestern University
Luciano Pomatto: California Institute of Technology

Sankhya A: The Indian Journal of Statistics, 2020, vol. 82, issue 1, No 1, 12 pages

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

Keywords: Finitely additive probabilities; Measure theory; Measurability; Primary 28A25; Secondary 60F15 (search for similar items in EconPapers)
Date: 2020
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