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

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 -