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**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 \(A\subseteq X^{\infty}\).

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

### MSC:

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 |