Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product

Ayed. R. A. Alanzi, Shokrya S. Alshqaq 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
Shokrya S. Alshqaq: Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia

Mathematics, 2024, vol. 12, issue 22, 1-9

Abstract: Let F ( ν j ) = { Q m j ν j , m j ∈ ( m − ν j , m + ν j ) } , j = 1 , 2 , be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν 1 and ν 2 . Introduce the set of measures F = F ( ν 1 ) ⊞ F ( ν 2 ) = { Q m 1 ν 1 ⊞ Q m 2 ν 2 , m 1 ∈ ( m − ν 1 , m + ν 1 ) a n d m 2 ∈ ( m − ν 2 , m + ν 2 ) } . We show that if F remains a CSK family, (i.e., F = F ( μ ) where μ is a non-degenerate compactly supported measure), then the measures μ , ν 1 and ν 2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F ) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F ) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

Keywords: variance function; free and Boolean convolutions; 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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