Studies on the Marchenko–Pastur Law

Ayed. R. A. Alanzi, Ohud A. Alqasem, Maysaa Elmahi Abd Elwahab and Raouf Fakhfakh ()
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Ayed. R. A. Alanzi: Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia
Ohud A. Alqasem: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Maysaa Elmahi Abd Elwahab: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia

Mathematics, 2024, vol. 12, issue 13, 1-10

Abstract: In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t -deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K + ( μ ) = { Q m μ ( d x ) ; m ∈ ( m 0 μ , m + μ ) } be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Q m μ • s the Fermi convolution power of order s > 0 of the measure Q m μ . We prove that if Q m μ • s ∈ K + ( μ ) , then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t -deformed free convolution t .

Keywords: variance function; Cauchy–Stieltjes transform; Marchenko–Pastur law (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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