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A characterization of the Marchenko–Pastur probability measure

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  • Fakhfakh, Raouf

Abstract

Consider K+(ν)={Q(m,ν)(dx);m∈(m0(ν),m+(ν))} the Cauchy-Stieltjes Kernel (CSK) family generated by a non degenerate probability measure ν with support bounded from above. Let X1,X2,…,Xn be boolean independent random variables with common distribution an element from K+(ν). i.e X1∼Q(m,ν) for some m∈(m0(ν),m+(ν)). We prove that the distribution of ∑i=1nXi belongs to the CSK family K+(ν) if and only if the probability measure ν is a scale transformation of the Marchenko–Pastur probability measure.

Suggested Citation

  • Fakhfakh, Raouf, 2022. "A characterization of the Marchenko–Pastur probability measure," Statistics & Probability Letters, Elsevier, vol. 191(C).
    • Handle: RePEc:eee:stapro:v:191:y:2022:i:c:s016771522200178x
    DOI: 10.1016/j.spl.2022.109660
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    References listed on IDEAS

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    1. Włodzimierz Bryc & Abdelhamid Hassairi, 2011. "One-Sided Cauchy–Stieltjes Kernel Families," Journal of Theoretical Probability, Springer, vol. 24(2), pages 577-594, June.
    2. Fakhfakh, Raouf, 2020. "Variance function of boolean additive convolution," Statistics & Probability Letters, Elsevier, vol. 163(C).
    3. 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.
    4. Fakhfakh, Raouf, 2017. "The mean of the reciprocal in a Cauchy–Stieltjes family," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 1-11.
    5. 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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