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Variance function of boolean additive convolution

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  • Fakhfakh, Raouf

Abstract

Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formula is used to identify the relation between variance functions under Boolean Bercovici–Pata Bijection between probability measures. We also give the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers.

Suggested Citation

  • Fakhfakh, Raouf, 2020. "Variance function of boolean additive convolution," Statistics & Probability Letters, Elsevier, vol. 163(C).
    • Handle: RePEc:eee:stapro:v:163:y:2020:i:c:s0167715220300808
    DOI: 10.1016/j.spl.2020.108777
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    References listed on IDEAS

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    1. Fakhfakh, Raouf, 2017. "The mean of the reciprocal in a Cauchy–Stieltjes family," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 1-11.
    2. Bryc, Włodek & Fakhfakh, Raouf & Hassairi, Abdelhamid, 2014. "On Cauchy–Stieltjes kernel families," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 295-312.
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    Cited by:

    1. Fakhfakh, Raouf, 2022. "A characterization of the Marchenko–Pastur probability measure," Statistics & Probability Letters, Elsevier, vol. 191(C).

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    3. Fakhfakh, Raouf, 2017. "The mean of the reciprocal in a Cauchy–Stieltjes family," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 1-11.

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