Let $\Gamma$ be a Borel probability measure on $\mathbb{R}$ and $(T,\mathcal{C},Q)$ a nonatomic probability space. Define $\mathcal{H}=\{H\in\mathcal{C}:Q(H)>0\}$. In some economic models, the following condition is requested. There are a probability space $(\Omega,\mathcal{A},P)$ and a real process $X=\{X_t:t\in T\}$ satisfying \begin{gather*} \text{for each }H\in\mathcal{H},\text{ there is }A_H\in\mathcal{A}\text{ with }P(A_H)=1\text{ such that } \\t\mapsto X(t,\omega)\text{ is measurable and }\,Q\bigl(\{t:X(t,\omega)\in\cdot\}\mid H\bigr)=\Gamma(\cdot)\,\text{ for }\omega\in A_H. \end{gather*} Such a condition fails if $P$ is countably additive, $\mathcal{C}$ countably generated and $\Gamma$ non trivial. Instead, as shown in this note, it holds for any $\mathcal{C}$ and $\Gamma$ under a finitely additive probability $P$. Also, $X$ can be taken to have any given distribution.
A note on the absurd law of large numbers in economics
RIGO, PIETRO
2012-01-01
Abstract
Let $\Gamma$ be a Borel probability measure on $\mathbb{R}$ and $(T,\mathcal{C},Q)$ a nonatomic probability space. Define $\mathcal{H}=\{H\in\mathcal{C}:Q(H)>0\}$. In some economic models, the following condition is requested. There are a probability space $(\Omega,\mathcal{A},P)$ and a real process $X=\{X_t:t\in T\}$ satisfying \begin{gather*} \text{for each }H\in\mathcal{H},\text{ there is }A_H\in\mathcal{A}\text{ with }P(A_H)=1\text{ such that } \\t\mapsto X(t,\omega)\text{ is measurable and }\,Q\bigl(\{t:X(t,\omega)\in\cdot\}\mid H\bigr)=\Gamma(\cdot)\,\text{ for }\omega\in A_H. \end{gather*} Such a condition fails if $P$ is countably additive, $\mathcal{C}$ countably generated and $\Gamma$ non trivial. Instead, as shown in this note, it holds for any $\mathcal{C}$ and $\Gamma$ under a finitely additive probability $P$. Also, $X$ can be taken to have any given distribution.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
