A note on the absurd law of large numbers in economics

作者：Berti Patrizia; Gori Michele; Rigo Pietro^*

来源：Journal of Mathematical Analysis and Applications, 2012, 388(1): 98-101.

DOI：10.1016/j.jmaa.2011.10.040

• Aggregate uncertainty
• Extension
• Finitely additive probability measure
• Individual risk
• Law of large numbers

Let Gamma be a Borel probability measure on R and (T, C, Q) a nonatomic probability space. Define H = {H is an element of C: Q(H) %26gt; 0}. In some economic models, the following condition is requested. There is a probability space (Omega, A, P) and a real process X = {X(t): t is an element of T} satisfying for each H is an element of H, there is A(H) is an element of A with P(A(H)) = 1 such that %26lt;br%26gt;t bar right arrow X(t, omega) is measurable and Q ({t: X(t, omega) is an element of . } vertical bar H) = Gamma(.) for omega is an element of A(H). %26lt;br%26gt;Such a condition fails if P is countably additive, C countably generated and Gamma nontrivial. Instead, as shown in this note, it holds for any C and Gamma under a finitely additive probability P. Also, X can be taken to have any given distribution.

• 出版日期2012-4-1