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Characterization of risk: a sharp law of large numbers

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Hammond, Peter J. 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 (No.806).

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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 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.

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

Official Date:

2007

Dates:

Date	Event
2007	Published

Number:

No.806

Number of Pages:

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Not Peer Reviewed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. Al-Najjar, Decomposition and characterization of risk with a continuum of random variables, Econometrica 63 (1995), 1195–1224.
[3] N. I. Al-Najjar, 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. Hoffmann-Jørgensen, The law of large numbers for non-measurable and non-separable 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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