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August, 1985 De Finetti-type Theorems: An Analytical Approach
Paul Ressel
Ann. Probab. 13(3): 898-922 (August, 1985). DOI: 10.1214/aop/1176992913


A famous theorem of De Finetti (1931) shows that an exchangeable sequence of $\{0, 1\}$-valued random variables is a unique mixture of coin tossing processes. Many generalizations of this result have been found; Hewitt and Savage (1955) for example extended De Finetti's theorem to arbitrary compact state spaces (instead of just $\{0, 1\}$). Another type of question arises naturally in this context. How can mixtures of independent and identically distributed random sequences with certain specified (say normal, Poisson, or exponential) distributions be characterized among all exchangeable sequences? We present a general theorem from which the "abstract" theorem of Hewitt and Savage as well as many "concrete" results--as just mentioned--can be easily deduced. Our main tools are some rather recent results from harmonic analysis on abelian semigroups.


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Paul Ressel. "De Finetti-type Theorems: An Analytical Approach." Ann. Probab. 13 (3) 898 - 922, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0579.60012
MathSciNet: MR799427
Digital Object Identifier: 10.1214/aop/1176992913

Primary: 60E05
Secondary: 43A35 , 44A05 , 60B99 , 62A05

Keywords: (completely) positive definite functions on $\ast$-semigroups , convolution semigroups , De Finetti's theorem , exchangeability , Hewitt and Savage's theorem , integrated Cauchy functional equation , multivariate survival function , Radon-presentability , Schoenberg triples

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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