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

• Autorzy: Pruss A. R.

Identyfikatory

DOI

10.4064/ba61-2-10

• Języki publikacji: EN

Abstrakty

EN

Let Ω be a countable infinite product Ω₁N 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 X_n(ω)=Ψ(Ξn(ω)). Then we can think of {X_n} as a sequence of independent but possibly nonmeasurable random variables on Ω. Let S_n=X₁+⋯+X_n. By the ordinary Strong Law of Large Numbers, we almost surely have E∗[X1]≤lim infSn/n≤lim supSn/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 S_n/n converges is maximally nonmeasurable: it has inner measure zero and outer measure one.

Słowa kluczowe

EN

law of large numbers; measurability; nonmeasurable random variables; probability

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2013
• Tom: Vol. 61, no 2
• Strony: 161--168
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Pruss A. R.

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

Bibliografia

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• [4] J. Hoffmann-Jørgensen, Perfect independence and stochastic inequalities, MaPhySto, Research Report, Dept. Math. Sci., Univ. of Aarhus, 2002, http://www.maphysto.dk/cgi-bin/gp.cgi?publ=375.
• [5] J. N. McDonald and N. A. Weiss, A Course in Real Analysis, Academic Press, San Diego, 1999.
• [6] A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics, Springer, New York, 1996.