One-Sided Cauchy–Stieltjes Kernel Families

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 illustra...

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Bibliographic Details
Published in:Journal of theoretical probability 2011-06, Vol.24 (2), p.577-594
Main Authors: Bryc, Włodzimierz, Hassairi, Abdelhamid
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
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Summary: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.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-010-0303-x