Infinitely divisible probability measures and potential kernels

1979 ◽  
pp. 22-35 ◽  
Author(s):  
Christian Berg ◽  
Gunnar Forst
Keyword(s):  
Probability Measures ◽  
Infinitely Divisible

scholarly journals The arithmetic of a semigroup of series of Walsh functions

2000 ◽  
Vol 68 (3) ◽  
pp. 365-378 ◽  
Author(s):  
I. P. Il'inskaya
Keyword(s):  
Real Line ◽  
Classical System ◽  
Convolution Semigroup ◽  
Probability Measures ◽  
Walsh Functions ◽  
Multiplicative Semigroup ◽  
Infinitely Divisible ◽  
The Real ◽  
Walsh Group ◽  
Series Of Functions

AbstractLet be the classical system of the Walsh functions, the multiplicative semigroup of the functions represented by series of functions Wk(t)with non-negative coefficients which sum equals 1. We study the arithmetic of . The analogues of the well-known [ related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in . The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: is isomorphic to the semigroup of probability measures on the groups of characters of the Cantor-Walsh group.

scholarly journals Infinitely divisible probability measures on a discrete Gelfand pair

1989 ◽  
Vol 116 ◽  
pp. 43-62 ◽  
Author(s):  
H. Heyer
Keyword(s):  
Double Coset ◽  
Direct Access ◽  
Probability Measures ◽  
Gelfand Pairs ◽  
Infinitely Divisible ◽  
Unpublished Thesis ◽  
Coset Spaces ◽  
Poisson Representation ◽  
Object Of Study ◽  
Discrete Spaces

Infinitely divisible probability measures on discrete spaces admitting a commutative convolution have been studied in various frameworks. For generalized convolutions related to Delphic structures an important contribution was made by Gilewski and Urbanik in [5]. In [11] Schwartz based his detailed analysis on convolutions arising from orthogonal series. Both of these approaches can be included into the framework of discrete hypergroups chosen f.e. by Gallardo and Gebuhrer in [4]. The main result common to these sources is the fact that, roughly speaking, all infinitely divisible probability measures are of Poisson type. Although the double coset spaces of Gelfand pairs are commutative hypergroups admitting an extended harmonic anafysis, the analytic methods developed in the theory of Gelfand pairs (see Dieudonné [3] and Heyer [7]) provide a more direct access to specific results like the characterization of divisible (idempotent, infinitely divisible) measures. For discrete Gelfand pairs (and their double coset spaces) Letac and his school have achieved remarkable results (see Letac [8], [9] and the references therein). The still unpublished thesis [1] of S. Ben Mansoor contains much information about divisible probability measures on cubes. In our exposition we reprove Ben Mansoor’s main theorem 3.5.4 with additional care, and discuss the special case of the ra-dimensional cube which was the basic object of study already in Letac, Takács [10]. It turns out that the main obstacle to be overcome in establishing the Poisson representation of infinitely divisible probability measures is the presence of idempotent factors, a problem that has been excluded f.e. in the work [4] of Gallardo and Gebuhrer.

Infinitely divisible probabilities on SL(2,) are continuously embedded

1982 ◽  
Vol 92 (1) ◽  
pp. 101-107 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Weak Topology ◽  
Probability Measures ◽  
Infinitely Divisible ◽  
Locally Compact

For any locally compact Hausdorff topological group G, let M(G) denote the topo-logical semigroup of all probability measures on G, where M (G) is given the weak topology and the multiplication is convolution.

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