Studies on the Marchenko–Pastur Law

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)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•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.