## 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 $$A\subseteq X^{\infty}$$.