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One-Sided Cauchy–Stieltjes Kernel Families

Author

Listed:
  • Włodzimierz Bryc

    (University of Cincinnati)

  • Abdelhamid Hassairi

    (Sfax University)

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.

Suggested Citation

  • Włodzimierz Bryc & Abdelhamid Hassairi, 2011. "One-Sided Cauchy–Stieltjes Kernel Families," Journal of Theoretical Probability, Springer, vol. 24(2), pages 577-594, June.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:2:d:10.1007_s10959-010-0303-x
    DOI: 10.1007/s10959-010-0303-x
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    Citations

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    Cited by:

    1. Bryc, Włodek & Fakhfakh, Raouf & Hassairi, Abdelhamid, 2014. "On Cauchy–Stieltjes kernel families," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 295-312.
    2. Fakhfakh, Raouf, 2017. "The mean of the reciprocal in a Cauchy–Stieltjes family," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 1-11.
    3. Fakhfakh, Raouf, 2022. "A characterization of the Marchenko–Pastur probability measure," Statistics & Probability Letters, Elsevier, vol. 191(C).
    4. Raouf Fakhfakh, 2021. "On some properties of Cauchy-Stieltjes Kernel families," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(4), pages 1186-1200, December.
    5. Ayed. R. A. Alanzi & Ohud A. Alqasem & Maysaa Elmahi Abd Elwahab & Raouf Fakhfakh, 2024. "Studies on the Marchenko–Pastur Law," Mathematics, MDPI, vol. 12(13), pages 1-10, July.

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