Classical theorems of probability on Gelfand pairs -- Khinchin's theorem and Cramér's theorem
type: Item
instance of: scholarly article
title: Classical theorems of probability on Gelfand pairs -- Khinchin's theorem and Cramér's theorem
zbMATH Open document ID: 1032.43001
DOI: 10.1007/BF02784506
published in: Israel Journal of Mathematics
publication date: 2003-03-13T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 1874388
zbMATH Keywords: Gelfand pairs
zbMATH Keywords: factorisation
zbMATH Keywords: Hungarian semigroup
zbMATH Keywords: Gaussian law
zbMATH Keywords: embedding problem
author: Piotr Graczyk
author: C. R. E. Raja
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
cites work: Harmonic analysis of probability measures on hypergroups
cites work: On Gelfand Pairs Associated with Solvable Lie Groups
cites work: Q4041031
cites work: Factors, roots and embeddability of measures on Lie groups
cites work: Q4934469
cites work: Représentations de semi-groupes de mesures sur un groupe localement compact
cites work: On shifted convolution powers of a probability measure
cites work: Q5288960
cites work: Distances hilbertiennes invariantes sur un espace homogene
cites work: Q3829998
cites work: Q4290905
cites work: Cramér theorem on symmetric spaces of noncompact type
cites work: Factorization theorems on symmetric spaces of noncompact type
cites work: Convolution semigroups of probability measures on Gelfand pairs
cites work: Infinitely divisible probability measures on a discrete Gelfand pair
cites work: Q4214023
cites work: Q5339349
cites work: Q5545096
cites work: Q3914127
cites work: Q5778668
cites work: Q3840391
cites work: Harmonic analysis on a class of generalized Gel'fand pairs associated with hyperbolic spaces
cites work: Q5533878
cites work: On a class of Hungarian semigroups and the factorization theorem of Khinchin
cites work: Q4040358
cites work: Semistable measures and limit theorems on real and \(p\)-adic groups
cites work: The central limit problem on locally compact groups
cites work: Limit theorems for probability measures on totally disconnected groups
cites work: A problem related to the arithmetic of probability measures on the sphere
cites work: Q4527707
cites work: Q3994337
cites work: On the heredity of Hun and Hungarian property
cites work: Q3687381
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description: scientific article; zbMATH DE number 1874388