An Abstract Law of Large Numbers

Nabil I. Al-Najjar; Luciano Pomatto

doi:10.1007/s13171-018-00162-z

An Abstract Law of Large Numbers

Nabil I. Al-Najjar, Luciano Pomatto^*

^*Corresponding author for this work

Managerial Economics, Decision Sciences and Operations

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 ∫ _IZ_idν(i). We establish that any ν that guarantees the measurability of ∫ _IZ_idν(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 ∫ _IZ_idν(i) equals the average expectation ∫ _IE[Z_i] dν(i).

Original language	English (US)
Pages (from-to)	1-12
Number of pages	12
Journal	Sankhya A
Volume	82
Issue number	1
DOIs	https://doi.org/10.1007/s13171-018-00162-z
State	Published - 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

Access to Document

10.1007/s13171-018-00162-z

Cite this

@article{db21a98f2a5a4db6adb2399167eb1516,

title = "An Abstract Law of Large Numbers",

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).",

keywords = "Finitely additive probabilities, Measurability, Measure theory, Primary 28A25, Secondary 60F15",

author = "Al-Najjar, {Nabil I.} and Luciano Pomatto",

note = "Publisher Copyright: {\textcopyright} 2019, Indian Statistical Institute.",

year = "2020",

month = feb,

day = "1",

doi = "10.1007/s13171-018-00162-z",

language = "English (US)",

volume = "82",

pages = "1--12",

journal = "Sankhya A",

issn = "0976-836X",

publisher = "Springer India",

number = "1",

}

TY - JOUR

T1 - An Abstract Law of Large Numbers

AU - Al-Najjar, Nabil I.

AU - Pomatto, Luciano

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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).

AB - 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).

KW - Finitely additive probabilities

KW - Measurability

KW - Measure theory

KW - Primary 28A25

KW - Secondary 60F15

UR - http://www.scopus.com/inward/record.url?scp=85060199126&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060199126&partnerID=8YFLogxK

U2 - 10.1007/s13171-018-00162-z

DO - 10.1007/s13171-018-00162-z

M3 - Article

AN - SCOPUS:85060199126

SN - 0976-836X

VL - 82

SP - 1

EP - 12

JO - Sankhya A

JF - Sankhya A

IS - 1

ER -

An Abstract Law of Large Numbers

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this