Characterization of Risk: A Sharp Law of Large Numbers

An extensive literature in economics uses a continuum of random variables to model individual random shocks imposed on a large population. Let H denote the Hilbert space of square-integrable random variables. A key concern is to characterize the family of all H-valued functions that satisfy the law of large numbers when a large sample of agents is drawn at random. We use the iterative extension of an infinite product measure introduced in [6] to formulate a “sharp” law of large numbers. We prove that an H-valued function satisfies this law if and only if it is both Pettis-integrable and norm integrably bounded. ∗Department of Economics, University of Warwick, Coventry CV4 7AL, U.K. e-mail: [email protected] †Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570. e-mail: [email protected]