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In the present paper, we investigate the model of a single-server queueing system with unlimited queue (of M/X/G/1/∞-type), random volume customers, unlimited memory buffer and multiple vacations. In analyzed system, arriving customers (that form Poisson entrance flow of groups of customers) transport some information measured in bytes so they are assumed to be additionally characterized by some non-negative random volume. Customer service time generally depends on his volume. Information delivered by a customer is written out into dedicated unlimited memory buffer until customer ends his service. In addition, in considered system the mechanism of multiple vacations is implemented which means that server can have some breaks (rests) for a random period of time but breaks begin only in the moments when there is no customer present in the system. The above-mentioned mechanism has obvious influence on customer waiting time and, in consequence, on customers total volume. For the introduced model, we obtain general formula for the steady-state customers total volume distribution in the terms of Laplace--Stieltjes transforms as well as formulae defining its first two moments. Analysis of some interesting, practical special cases of the model and numerical computations are attached as well together with examples of possible applications of the model regarding real telecommunication or computer systems. [ABSTRACT FROM AUTHOR]

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function F α , β s by using two different approaches: the Bromwich integral and the decomposition of F α , β s in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms. [ABSTRACT FROM AUTHOR]

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly. [ABSTRACT FROM AUTHOR]

In this work, we establish some Parseval-Goldstein type identities and relations that include various new generalized integral transforms such as Lα,µ-transform and generalized Stieltjes transform. In addition, we evaluated improper integrals of some fundamental and special functions using our results. [ABSTRACT FROM AUTHOR]

In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in Zhou (JTP 36:1203–1226, 2023). Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in Zhou (JTP 36:1203-1226, 2023) to allow for all possible ratios of row to column of the underlying random matrix. [ABSTRACT FROM AUTHOR]

In this paper, we explore the properties of the generalized Lambert transform, the L-transform, the generalized Stieltjes transform, and the Stieltjes–Poisson transform within the framework of Lebesgue spaces. We establish Parseval-type relations for each transform, providing a comprehensive analysis of their behaviour and mathematical characteristics. [ABSTRACT FROM AUTHOR]

We establish a new perturbation theory for orthogonal polynomials using a Riemann–Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization, and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms. [ABSTRACT FROM AUTHOR]

We study continuous-time open quantum walks in one dimension through a matrix representation, focusing on nearest-neighbor transitions for which an associated weight matrix exists. Statistics such as site recurrence are studied in terms of matrix-valued orthogonal polynomials and explicit calculations are obtained for classes of Lindblad generators that model quantum versions of birth-death processes. Emphasis is given to the technical distinction between the cases of a finite or infinite number of vertices. Recent results for open quantum walks are adapted in order to apply the folding trick to continuous-time birth-death chains on the integers. Finally, we investigate the matrix-valued Stieltjes transform associated to the weights. [ABSTRACT FROM AUTHOR]

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function Fα,βs by using two different approaches: the Bromwich integral and the decomposition of Fα,βs in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms.

We implement a complex analytic method to build an estimator of the spectrum of a matrix perturbed by the addition of a random matrix noise in the free probabilistic regime. This method, which has been previously introduced by Arizmendi, Tarrago and Vargas, involves two steps: the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. This method thus reduces the spectral deconvolution problem to a classical one. We provide explicit bounds for the mean squared error of the first step under the assumption that the distribution of the noise is unitary invariant. In the case where the unknown measure is sparse or close to a distribution with a density with enough smoothness, we prove that the resulting estimator converges to the measure in the |$1$| -Wasserstein distance at speed |$O(1/\sqrt{N})$|⁠ , where |$N$| is the dimension of the matrix. [ABSTRACT FROM AUTHOR]

Consider an N by N matrix X of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of X*X converges to the Marchenko–Pastur law on the optimal scale with probability 1. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under the assumption that the entries have a finite fourth moment and are truncated at N1/4, or alternatively with exploding moments. In this work we simplify and adapt the methods from prior papers of Götze–Tikhomirov [Probab. Relat. Fields 165(1–2), 163–233 (2016)] and Cacciapuoti–Maltsev–Schlein [Probab. Theory Relat. Fields 163(1–2), 1–59 (2015)] to covariance matrices. [ABSTRACT FROM AUTHOR]

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below. [ABSTRACT FROM AUTHOR]

Let \omega and \nu be radial weights on the unit disc of the complex plane such that \omega admits the doubling property \sup _{0\le r<1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}<\infty. Consider the one weight inequality \begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1

In this paper, we deal with the iteration of the Laplace transform over certain spaces of generalized functions. As a consequence we prove a new Post-Widder-type inversion formula for this transform over distributions of compact support. [ABSTRACT FROM AUTHOR]

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions  E α , α ∈ (0 , 2 ] , enjoy the  HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable  α and complex numbers z, the real and imaginary part of the functions  x ↦ E α z x , are tightly linked to the stable distributions and to the generalized Cauchy kernel. [ABSTRACT FROM AUTHOR]

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W 0 (t) , in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions. [ABSTRACT FROM AUTHOR]

M. Loeve wrote that "the fundamental limit theorems of Probability theory may be classified into two groups. One group deals with the problem of limit laws of sequences of some of random variables, the other deals with the problem of limits of random variables, in the sense of almost sure convergence, of such sequences. The list of topics is extensive, and it includes classical models of sums of both independent and various types of dependent random variables, probabilities of large deviations, functional limit theorems, and limit theorems for random processes, in high-dimensional spaces, for spectra of random matrices and random graphs, and more. It is assumed that the covariance structure is one of three covariance structures: (1) an independent covariance structure with the same variance, (2) an independent covariance structure with different variances, and (3) a uniform covariance structure. [Extracted from the article]

In the present paper four theorems connecting Stieltjes transform and Hankel transform are established. The theorems are general in nature. Four integral formulae involving special functions are obtained with the help of these theorems. Otherwise it is very difficult to evaluate such type of integrals. Other several integrals may be evaluated with the help of these theorems.

Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-parameter family of Stieltjes transformations related to holomorphic Lambert–Tsallis functions, which are a two-parameter generalization of the Lambert function. Such Stieltjes transformations appear in the study of eigenvalue distributions of random matrices associated with some growing statistically sparse models. A necessary and sufficient condition on the parameters is given for the corresponding functions being Stieltjes transformations of probabilistic measures. We also give an explicit formula of the corresponding R-transformations. [ABSTRACT FROM AUTHOR]

Let X n be n × N random complex matrices, and let R n and T n be non-random complex matrices with dimensions n × N and n × n , respectively. We assume that the entries of X n are normalized independent random variables satisfying the Lindeberg condition, T n are nonnegative definite Hermitian matrices and commutative with R n R n ∗ , i.e., T n R n R n ∗ = R n R n ∗ T n . The general information-plus-noise-type matrices are defined by C n = 1 N T n 1 2 R n + X n R n + X n ∗ T n 1 2 . In this paper, we establish the limiting spectral distribution of the large-dimensional general information-plus-noise-type matrices C n . Specifically, we show that as n and N tend to infinity proportionally, the empirical distribution of the eigenvalues of C n converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform. [ABSTRACT FROM AUTHOR]

We extend integral transforms of Stieltjes type to a larger space of Boehmians than that used in previous studies. Our analysis includes the Stieltjes transform, the Poisson transform (Widder's potential transform) and the generalized Stieltjes transform. [ABSTRACT FROM AUTHOR]

Our goal in this article is to derive Abelian theorems for the two-sided Laplace transform, Mellin transform, one-sided real Laplace transform and Stieltjes transform over distributions of compact support and over certain function spaces of generalized functions. [ABSTRACT FROM AUTHOR]

In the present paper four theorems connecting Stieltjes transform and Hankel transform are established. The theorems are general in nature. Four integral formulae involving special functions are obtained with the help of these theorems. Otherwise it is very difficult to evaluate such type of integrals. Other several integrals may be evaluated with the help of these theorems. [ABSTRACT FROM AUTHOR]

By making use of the basic properties of Laplace-Stieltjes transform, we establish some inequalities concerning the abscissa of convergence, the abscissa of absolute convergence and the abscissa of uniform convergence of Laplace-Stieltjes transform ƒ∞0 estdα(t). Moreover, we point out that our formulas for the three abscissas of convergence are consistent with the previous results given by Yu [26] for Laplace-Stieltjes transform under some conditions. [ABSTRACT FROM AUTHOR]

We consider the limit of the empirical spectral distribution of Laplace matrices of generalized random graphs. Applying the Stieltjes transform method, we prove under general conditions that the limit spectral distribution of Laplace matrices converges to the free convolution of the semicircular law and the normal law. [ABSTRACT FROM AUTHOR]

Fisher matrix is one of the most important statistics in multivariate statistical analysis. Its eigenvalues are of primary importance for many applications, such as testing the equality of mean vectors, testing the equality of covariance matrices and signal detection problems. In this paper, we establish the limiting spectral distribution of high-dimensional noncentral Fisher matrices and investigate its analytic behavior. In particular, we show the determination criterion for the support of the limiting spectral distribution of the noncentral Fisher matrices, which is the base of investigating the high-dimensional problems concerned with noncentral Fisher matrices. [ABSTRACT FROM AUTHOR]

We focus on kernel estimator of the density function of the limiting spectral distribution of sample covariance matrix associated firstly, to a large class of weak dependent sequences of realvalued random variables having only moment of order 2. Afterwards, sample covariance matrices of the block-independent model and the random tensor model, where the data does not have independent coordinates. In the last two cases, we show that the kernel estimator of the density function converges to the Marchenko-Pastur density with probability one. A simulation study is conducted to show the performance of the kernel estimators of the density function and then compare these estimators with the one obtained by the Stieltjes transform method. [ABSTRACT FROM AUTHOR]

In the present paper, the integral transform which will be called generalized Laplace transform is considered. Iteration of generalized Laplace transform and Fourier cosine transform and Fourier sine transform is generalized Glasser transform. Some identities involving these transforms and the Mellin transform and the generalized Stieltjes transform are given. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to parametrize the Weyl matrix balls associated with an arbitrary matricial truncated Hamburger moment problem. For the special case of a non-degenerate matricial truncated Hamburger moment problem, the corresponding Weyl matrix balls were computed by I. V. Kovalishina [34] in the framework of V. P. Potapov's method of 'Fundamental matrix inequalities'. [ABSTRACT FROM AUTHOR]

• The notion of an asymptotic continued fraction expansion of a complex-valued function is introduced. • Some general results about asymptotic continued fraction expansions are established. • Two continued fraction expansions of the exponential integral E n (z) are shown to yield two asymptotic continued fraction expansions of the prime counting function π (x). • All of the best rational function approximations of the function π (e x) / e x are determined. • Our results on π (x) are generalized to any arithmetic semigroup satisfying Axiom A, hence to any number field. We provide several asymptotic expansions of the prime counting function π (x) and related functions. We define an asymptotic continued fraction expansion of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer n , two well-known continued fraction expansions of the exponential integral function E n (z) correspondingly yield two asymptotic continued fraction expansions of π (x) / x. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the "best" rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function π (e x) / e x. Finally, we generalize our results on π (x) to any arithmetic semigroup satisfying Axiom A, and thus to any number field. [ABSTRACT FROM AUTHOR]

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W0(t), in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions.

Vinberg cones and the ambient vector spaces are important in modern statistics of sparse models. The aim of this paper is to study eigenvalue distributions of Gaussian, Wigner and covariance matrices related to growing Vinberg matrices. For Gaussian or Wigner ensembles, we give an explicit formula for the limiting distribution. For Wishart ensembles defined naturally on Vinberg cones, their limiting Stieltjes transforms, support and atom at 0 are described explicitly in terms of the Lambert–Tsallis functions, which are defined by using the Tsallis q-exponential functions. [ABSTRACT FROM AUTHOR]

For b > 1 and αβ a string of two digits in base b , let K 1 be the subsum of the harmonic series with only those integers having exactly one occurrence of αβ. We obtain a theoretical representation of such K 1 series which, say for b = 10 , allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the K 1 sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation. [ABSTRACT FROM AUTHOR]

This article is concerned with the spectral behavior of p-dimensional linear processes in the moderately high-dimensional case when both dimensionality p and sample size n tend to infinity so that p/n→0. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of pdimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the p × p linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

B Book review b Graph © 2021, Cambridge University Press Random matrix theory (RMT) was first developed in the physics community as an approximation to understand large systems in quantum mechanics. It is exhaustive enough (including recent developments) to appeal to seasoned research scientists that are looking for hands-on RMT techniques and computations to be applied to their own particular data problems. It is central in the modern RMT as it is well suited to understand the universality of the eigenvalue distribution as well as the effect of adding two random matrices together. [Extracted from the article]

We study the empirical spectral distributions of two sample-covariance-type matrices associated with high-dimensional time series with unit roots. The first matrix is S = XX'/T, where X is an n×T data set, with rows represented by n independent and identically distributed (i.i.d.) copies of T consecutive observations of a difference-stationary process. The second matrix is W = n∫R0¹ Wn(t)Wn(t)0 dt, where Wn(t) is an n-dimensional vector with i.i.d. Brownian motion components. We show that as n and T diverge to infinity proportionally, the two distributions weakly converge to non-random limits. The limit corresponding to S has a density φ(x) that decays as x-3/2 when x →8734: The limit corresponding to W is a Feller-Pareto distribution. An illustrative application is provided. [ABSTRACT FROM AUTHOR]

We consider the limit of the empirical spectral distribution of Laplace matrices of generalized random graphs. Applying the Stieltjes transform method, we prove under general conditions that the limit spectral distribution of Laplace matrices converges to the free convolution of the semicircular law and the normal law.

Wiener–Hopf Equations are of the form u (t) = f (t) + ∫ 0 + ∞ k (t − τ) u (τ) d τ , t > 0. These equations arise in many physical problems such as radiative transport theory, reflection of an electromagnetive plane wave, sound wave transmission from a tube, and in material science. They are also known as the renewal equations on the half-line in Probability Theory. In this paper, we present a method of deriving asymptotic expansions for the solutions to these equations. Our method makes use of the Wiener–Hopf technique as well as the asymptotic expansions of Stieltjes and Hilbert transforms. [ABSTRACT FROM AUTHOR]

We consider sparse sample covariance matrices with sparsity probability with . Assuming that the distribution of matrix elements has a finite absolute moment of order , , it is shown that the distance between the Stieltjes transforms of the empirical spectral distribution function and the Marchenko–Pastur law is of order , where v is the distance to the real axis in the complex plane. [ABSTRACT FROM AUTHOR]

We study the eigenvalue distribution of a Gaussian unitary ensemble (GUE) matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu's notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results. Deformed random matrix, Variance profile, Outlier detection, Free probability, Freeness with amalgamation, Operator-valued Stieltjes transform, Gaussian spiked model, Low-rank model. 2000 Math Subject Classification: 62G05, 62H12. [ABSTRACT FROM AUTHOR]

This paper is concerned with extensions of the classical Mařenko-Pastur law to time series. Specifically, p-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed (real-or complex-valued) entries possessing zero mean, unit variance and finite fourth moments. The coefficient matrices of the linear process are assumed to be simultaneously diagonalizable. In this setting, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting p/n→c ∈ (0,∞) for which dimension p and sample size n diverge to infinity at the same rate. The results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case.