De Finetti-type theorems: An analytical approach. (English) Zbl 0579.60012

In this paper the well known theorem of E. Hewitt and L. J. Savage [Trans. Am. Math. Soc. 80, 470-501 (1955; Zbl 0066.296)] is generalized and from this theorem a result of the type of De Finetti’s theorem is deduced. Let A be the family of all complex-valued measurable functions on some measurable space (X,B) which are bounded by 1. Let S denote an Abelian semigroup and let Z denote an abstract set with involution. Furthermore three mappings \(a: Z\to A\), \(t: Z\to S\), \(\beta\) : \(Z\to {\mathbb{C}}\setminus \{0\}\) are given such that \(a(z^*)=\overline{a(z)}\), \(\beta (z^*)=\overline{\beta (z)}\), \(t(z^*)=(t(z))^*\) and such that t(z) generates S.
Theorem 1. Under the assumptions just given, let \(X_ 1,X_ 2,...\), be a sequence of X-valued random variables such that \[ E[\prod^{n}_{j=1}a(z_ j)\circ X_ j]=\prod^{n}_{j=1}\beta (z_ j)\phi (\sum^{n}_{j=1}t(z_ j)) \] for all \(n\geq 1\) and all \(z_ 1,...,z_ n\in Z\), where \(\phi\) : \(S\to {\mathbb{C}}\) is some function normalized by \(\phi (0)=1\). Then \(\phi\) is exponentially bounded and positive definite. In case the functions in \(A_ 0:=a(Z)\) are nonnegative and \(\beta >0\), \(\phi\) is even completely positive definite.
Theorem 2. Let P denote an exchangeable Radon probability measure on the countable infinite product \(X^{\infty}\) of some compact Hausdorff space X. Then for some uniquely determined Radon probability measure \(\mu\) on \(M^ 1_+(X)\), the space of all Radon probabilities on X, we have \(P(A)=\int k^{\infty}(A)d\mu (k)\) for each Borel set \(A\subseteq X^{\infty}\).
Reviewer: G.A.Sokhadze


60E05 Probability distributions: general theory
43A35 Positive definite functions on groups, semigroups, etc.
44A05 General integral transforms
60B99 Probability theory on algebraic and topological structures
62A01 Foundations and philosophical topics in statistics


Zbl 0066.296
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