One-Sided Cauchy–Stieltjes Kernel Families

Włodzimierz Bryc () and Abdelhamid Hassairi ()
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Włodzimierz Bryc: University of Cincinnati
Abdelhamid Hassairi: Sfax University

Journal of Theoretical Probability, 2011, vol. 24, issue 2, 577-594

Abstract: Abstract This paper continues the study of a kernel family which uses the Cauchy–Stieltjes kernel 1/(1−θ x) in place of the celebrated exponential kernel exp (θ x) of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic pseudo-variance functions correspond to free-infinitely divisible laws without the first moment. We also determine the domain of means, advancing the understanding of Cauchy–Stieltjes kernel families also for compactly supported generating measures.

Keywords: Exponential families; Cauchy kernel; Free stable law; 60E10; 46L54 (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10959-010-0303-x

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