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Characterization of risk: a sharp law of large numbers
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Hammond, Peter J., 1945 and Sun, Yeneng (2007) Characterization of risk: a sharp law of large numbers. Working Paper. Coventry: University of Warwick, Department of Economics. (Warwick economic research papers).

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Official URL: http://www2.warwick.ac.uk/fac/soc/economics/resear...
Abstract
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 squareintegrable random variables. A key concern is to characterize the family of all Hvalued 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 Hvalued function satisfies this law if and only if it is both Pettisintegrable and norm integrably bounded.
Item Type:  Working or Discussion Paper (Working Paper) 

Subjects:  H Social Sciences > HB Economic Theory Q Science > QA Mathematics 
Divisions:  Faculty of Social Sciences > Economics 
Library of Congress Subject Headings (LCSH):  Hilbert space, Stochastic partial differential equations, Risk  Mathematical models, Law of large numbers, Mathematical statistics 
Series Name:  Warwick economic research papers 
Publisher:  University of Warwick, Department of Economics 
Place of Publication:  Coventry 
Date:  2007 
Number:  No.806 
Number of Pages:  8 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
References:  [1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, 2nd edition, Springer, New York, 1999. [2] N. I. AlNajjar, Decomposition and characterization of risk with a continuum of random variables, Econometrica 63 (1995), 1195–1224. [3] N. I. AlNajjar, Decomposition and characterization of risk with a continuum of random variables: corrigendum, Econometrica 67 (1999), 919–920. [4] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd edition, Springer, New York, 1997. [5] J. Diestel and J. J. Uhl, Vector Measures, American Mathematical Society, Rhode Island, 1977. [6] P. J. Hammond and Y. N. Sun, The essential equivalence of pairwise and mutual conditional independence, Probability Theory and Related Fields 135 (2006), 415–427. [7] P. J. Hammond and Y. N. Sun, Monte Carlo simulation of macroeconomic risk with a continuum of agents: The general case, Warwick Economics Research Paper No. 803 (2007) available from http://ideas.repec.org/p/wrk/warwec/803.html; Economic Theory, forthcoming. [8] J. HoffmannJørgensen, The law of large numbers for nonmeasurable and nonseparable random elements, Ast´erisque 131 (1985), 299–356. [9] M. A. Khan and Y. N. Sun, Weak measurability and characterizations of risk, Economic Theory 13 (1999), 541–560. 
URI:  http://wrap.warwick.ac.uk/id/eprint/1407 
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