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15 pages, 314 KiB

Open AccessArticle

by Shokrya S. Alshqaq, Raouf Fakhfakh and Fatimah Alshahrani

Axioms 2025, 14(6), 453; https://doi.org/10.3390/axioms14060453 - 9 Jun 2025

Viewed by 351

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free Gamma laws in the setting of Cauchy-Stieltjes Kernel (CSK) families. Our investigation reveals the essential links between analytic function theory and free probability, highlighting the usefulness of CSK families in developing the theoretical and computational aspects of free additive convolution. Full article

(This article belongs to the Section Mathematical Analysis)

13 pages, 292 KiB

Open AccessArticle

V_a-Deformed Free Convolution

by Fahad Alsharari and Raouf Fakhfakh

Mathematics 2025, 13(4), 572; https://doi.org/10.3390/math13040572 - 9 Feb 2025

Viewed by 565

Abstract

In this article, we study

V_{a}

-transformation of a measure and of a convolution (denoted by

a

) defined for

a \in R

. We provide significant insights into the stability of the free Meixner family of probability measures under

V_{a}

[...] Read more.

In this article, we study

V_{a}

-transformation of a measure and of a convolution (denoted by

a

) defined for

a \in R

. We provide significant insights into the stability of the free Meixner family of probability measures under

V_{a}

-transformation. We show that the

V_{a}

-transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. We also present some properties of the Marchenko–Pastur law in connection with

a

-convolution. In addition, some new limit theorems are proved for the

a

-convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to

V_{a}

-deformed free cumulants are presented. Full article

(This article belongs to the Section D1: Probability and Statistics)

9 pages, 328 KiB

Open AccessArticle

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

by Ayed. R. A. Alanzi, Shokrya S. Alshqaq and Raouf Fakhfakh

Mathematics 2024, 12(22), 3465; https://doi.org/10.3390/math12223465 - 6 Nov 2024

Cited by 1 | Viewed by 853

Abstract

Let

F (ν_{j}) = {Q_{m_{j}}^{ν_{j}}, m_{j} \in (m_{-}^{ν_{j}}, m_{+}^{ν_{j}})}

j = 1, 2

, be two Cauchy–Stieltjes Kernel (CSK) families induced [...] Read more.

Let

F (ν_{j}) = {Q_{m_{j}}^{ν_{j}}, m_{j} \in (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} \in (m_{-}^{ν_{1}}, m_{+}^{ν_{1}}) a n d m_{2} \in (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. Full article

(This article belongs to the Section D1: Probability and Statistics)

10 pages, 279 KiB

Open AccessArticle

Studies on the Marchenko–Pastur Law

by Ayed. R. A. Alanzi, Ohud A. Alqasem, Maysaa Elmahi Abd Elwahab and Raouf Fakhfakh

Mathematics 2024, 12(13), 2060; https://doi.org/10.3390/math12132060 - 1 Jul 2024

Viewed by 1297

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 \in (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} \in 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

\begin{matrix} t \end{matrix}

. Full article

(This article belongs to the Section D1: Probability and Statistics)

38 pages, 402 KiB

Open AccessArticle

Local Laws for Sparse Sample Covariance Matrices

by Alexander N. Tikhomirov and Dmitry A. Timushev

Mathematics 2022, 10(13), 2326; https://doi.org/10.3390/math10132326 - 3 Jul 2022

Cited by 2 | Viewed by 1693

Abstract

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order

n \times m

with

n / m \to y

(where

y > 0

) and sparse probability

n p_{n} > {log}^{β} n

[...] Read more.

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order

n \times m

with

n / m \to y

(where

y > 0

) and sparse probability

n p_{n} > {log}^{β} n

(where

β > 0

). The bounds of the distance between the empirical spectral distribution function of the sparse sample covariance matrices and the Marchenko–Pastur law distribution function that was obtained in the complex domain

z \in D

with

Im z > v_{0} > 0

(where

v_{0}

) were of order

{log}^{4} n / n

and the domain bounds did not depend on

p_{n}

while

n p_{n} > {log}^{β} n

. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

14 pages, 1057 KiB

Open AccessArticle

Signal Detection in Nearly Continuous Spectra and ℤ₂-Symmetry Breaking

by Vincent Lahoche, Dine Ousmane Samary and Mohamed Tamaazousti

Symmetry 2022, 14(3), 486; https://doi.org/10.3390/sym14030486 - 28 Feb 2022

Cited by 7 | Viewed by 2376

Abstract

The large scale behavior of systems having a large number of interacting degrees of freedom is suitably described using the renormalization group from non-Gaussian distributions. Renormalization group techniques used in physics are then expected to provide a complementary point of view on standard methods used in data science, especially for open issues. Signal detection and recognition for covariance matrices having nearly continuous spectra is currently an open issue in data science and machine learning. Using the field theoretical embedding introduced in Entropy, 23(9), 1132 to reproduce experimental correlations, we show in this paper that the presence of a signal may be characterized by a phase transition with

Z_{2}

-symmetry breaking. For our investigations, we use the nonperturbative renormalization group formalism, using a local potential approximation to construct an approximate solution of the flow. Moreover, we focus on the nearly continuous signal build as a perturbation of the Marchenko-Pastur law with many discrete spikes. Full article

(This article belongs to the Section Physics)

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