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

Alexander R. Pruss¹

¹ Department of Philosophy Baylor University One Bear Place #97273 Waco, TX 76798-7273, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) no. 2, pp. 161-168.

Résumé

Let $\varOmega$ be a countable infinite product $\varOmega_{1}^{\mathbb N}$ of copies of the same probability space $\varOmega_1$, and let $\{ \varXi_n \}$ be the sequence of the coordinate projection functions from $\varOmega$ to~$\varOmega_1$. Let $\varPsi$ be a possibly nonmeasurable function from $\varOmega_1$ to $\mathbb R$, and let $X_n(\omega) = \varPsi(\varXi_n(\omega))$. Then we can think of $\{ X_n \}$ as a sequence of independent but possibly nonmeasurable random variables on $\varOmega$. Let $S_n = X_1+\cdots+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \le \liminf S_n/n \le \limsup S_n/n \le E^*[X_1]$, where $E_*$ and $E^*$ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of $S_n/n$ in the nontrivial case where $E_*[X_1] E^*[X_1]$, and obtain several negative answers. For instance, the set of points of $\varOmega$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one.

Détail

DOI : 10.4064/ba61-2-10

Keywords: varomega countable infinite product varomega mathbb copies probability space varomega varxi sequence coordinate projection functions varomega varomega varpsi possibly nonmeasurable function varomega mathbb omega varpsi varxi omega think sequence independent possibly nonmeasurable random variables varomega cdots ordinary strong law large numbers almost surely have * liminf limsup * where * * lower upper expectations ask anything precise said about limit points nontrivial where * * obtain several negative answers instance set points varomega where converges maximally nonmeasurable has inner measure zero outer measure