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

Alexander R. Pruss

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013), p. 161-168 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1290.60038

EUDML-ID : urn:eudml:doc:281305