Classical theorems of probability on Gelfand pairs -- Khinchin's theorem and Cramér's theorem
The paper under review is concerned with 'classical' results on the arithmetic structure of probabilities due to Khinchin (for the real line), and their generalizations to Gelfand pairs. These 'classical' results are in particular \(\bullet\) \(\mu \in \mathcal{M}^1(\mathbb{R})\) is representable as convolution product of (at most countable) indecomposable factors and an anti-irreducible factor (i.e. without indecomposable factor), \(\bullet\) anti-irreducible probabilities are infinitely divisible, and \(\bullet\) limits of infinitesimal triangular arrays are infinitely divisible and vice versa. The investigations rely on (1) the structure of (certain) Gelfand pairs and hypergroups, (2) the theory of Hun and Hungarian semigroups [cf. e.g. \textit{I. Z. Ruzsa} and \textit{G. J. Szekely}, Algebraic Probability Theory (Wiley Series in Probability and Mathematical Statistics. Chichester (UK): Wiley. xiii, 251 p.) (1988; Zbl 0653.60012)] and (3) embedding of infinitely divisible probabilities into continuous convolution semigroups [see e.g. \textit{M. McCrudden}, An introduction to the embedding problem for probabilities on locally compact groups. In: Positivity in Lie Theory: Open Problems (Expositiones Math., de Gruyter) (1998; Zbl 0908.43002)]. They continue earlier investigations [see in particular the first named author, J. Theor. Probab. 7, 609-613 (1994; Zbl 0808.60013); the second author, J. Theor. Probab. 12, 561-569 (1999; Zbl 0934.60010) and \textit{R. Shah}, Isr. J. Math. 110, 189-218 (1999; Zbl 0943.60002)]. Section 2 contains a survey on Gelfand pairs \((G,K)\) satisfying a condition (*) which guarantees that the corresponding semigroup \(\mathcal{S}\) of \(K\)-biinvariant measures (resp. the hypergroup \(G//K\)) is Hungarian. Section 3 is concerned with factor-compactness of probabilities in \(\mathcal{S}\) and Section 4 contains the above-mentioned Khinchin's factorisation theorem for probabilities in \(\mathcal{S}\) if \(G\) is connected. Sections 5 and 6 contain analogous results for other types of Gelfand pairs, in particular for discrete \(G\), more generally, for \((G,K)\) possessing an open compact \(M\vartriangleleft G\), for \(2-\)root compact symmetric pairs and for Zariski-connected \(p-\)adic \(G\). Section 7 contains embedding results for \(\mathcal{S}\) on Gelfand pairs with strongly root compact normalizer \(N(K)\) and in Section 8 the existence of continuous partial homomorphisms -- a crucial part of the theory of Hungarian semigroups -- is proved for Abelian hypergroups, hence for \(\mathcal{S}\). Consequently, in Theorem 8.1 the authors obtain under the assumptions of the previous sections that Khinchin's three theorems hold true. Furthermore they obtain conditions under which anti-irreducible probabilities are continuously embeddable. Sections 9 and 10 are concerned with Gaussian laws. In particular, for compact connected \(G\) and a Gelfand pair \((G,K)\) Gaussian probabilities which are absolutely continuous with respect to the Haar measure do not obey Cramer's law: Gaussian probabilities may have indecomposable -- hence non-Gaussian -- factors.
scientific article; zbMATH DE number 1874388
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