On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss

Bulletin of the Polish Academy of Sciences. Mathematics (2013)

  • Volume: 61, Issue: 2, page 161-168
  • ISSN: 0239-7269

Abstract

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Let Ω be a countable infinite product Ω of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have E * [ X ] l i m i n f S / n l i m s u p S / n E * [ X ] , where E * and E* are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sₙ/n in the nontrivial case where E * [ X ] < E * [ X ] , and obtain several negative answers. For instance, the set of points of Ω where Sₙ/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.

How to cite

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Alexander R. Pruss. "On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables." Bulletin of the Polish Academy of Sciences. Mathematics 61.2 (2013): 161-168. <http://eudml.org/doc/281305>.

@article{AlexanderR2013,
abstract = {Let Ω be a countable infinite product $Ω₁^\{ℕ\}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_\{*\}[X₁] ≤ lim inf Sₙ/n ≤ lim sup Sₙ/n ≤ E*[X₁]$, where $E_\{*\}$ and E* are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sₙ/n in the nontrivial case where $E_\{*\}[X₁] < E*[X₁]$, and obtain several negative answers. For instance, the set of points of Ω where Sₙ/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.},
author = {Alexander R. Pruss},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {law of large numbers; measurability; nonmeasurable random variables; probability},
language = {eng},
number = {2},
pages = {161-168},
title = {On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables},
url = {http://eudml.org/doc/281305},
volume = {61},
year = {2013},
}

TY - JOUR
AU - Alexander R. Pruss
TI - On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2013
VL - 61
IS - 2
SP - 161
EP - 168
AB - Let Ω be a countable infinite product $Ω₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}[X₁] ≤ lim inf Sₙ/n ≤ lim sup Sₙ/n ≤ E*[X₁]$, where $E_{*}$ and E* are the lower and upper expectations. We ask if anything more precise can be said about the limit points of Sₙ/n in the nontrivial case where $E_{*}[X₁] < E*[X₁]$, and obtain several negative answers. For instance, the set of points of Ω where Sₙ/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.
LA - eng
KW - law of large numbers; measurability; nonmeasurable random variables; probability
UR - http://eudml.org/doc/281305
ER -

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