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

• Autorzy: Alexander R. Pruss
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 60F10: Large deviations
• 28A12: Contents, measures, outer measures, capacities
• 60F05: Central limit and other weak theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2013
• Tom: 61
• Numer: 2
• Strony: 161-168

Daty

wydano

2013

Twórcy

autor

Alexander R. Pruss

• Department of Philosophy, Baylor University, One Bear Place #97273, Waco, TX 76798-7273, U.S.A.

Identyfikatory

DOI

10.4064/ba61-2-10