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.
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Research partially supported by NSF grant DMS-0904720 and by Taft Research Seminar 2008/09.
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Bryc, W., Hassairi, A. One-Sided Cauchy–Stieltjes Kernel Families. J Theor Probab 24, 577–594 (2011). https://doi.org/10.1007/s10959-010-0303-x
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DOI: https://doi.org/10.1007/s10959-010-0303-x