arXiv subjects

Nomvelo Karabo Sibisi

Publications and source records attributed to Nomvelo Karabo Sibisi.

6 recordsLinked to original sources

arXiv2026

Representation-free Densities of Ratios and Powers of Stable Variables

We derive representation-free densities of ratios of the positive stable random variable and obtain their Laplace/Stieltjes transforms. We state representation-free densities of occupation times (Lamperti, 1958) and Bessel process excursions (Bertoin et al., 2006) and their transforms. We also obtain simple representations of these and other densities using a known infinite series representation of the stable density.

math.PR↗

arXiv2025

Distributions based on Stable Mixtures and Gamma-Stable Convolutions

This paper explores mixture distributions induced by a product of the positive stable random variable and a power of another positive random variable. The paper also considers the convolution of the stable density with a gamma density. These two constructs, mixing and convolution, suffice to generate a rich family of distributions. An example is the positive Linnik distribution, which is known to arise from a product involving stable and gamma random variables. We show that gamma-Linnik convolution gives the Mittag-Leffler distribution and the Mittag-Leffler Markov chain associated with the growth of random trees. Building on that, we construct a new family of distributions with explicit densities. A particular choice of parameters for this family yields the Lamperti-type laws associated with occupation times for Markov processes.

math.PR↗

arXiv2024

Convolutions and Mixtures of Gamma, Stable and Mittag-Leffler Distributions

This paper uses convolutions of the gamma density and the one-sided stable density to construct higher level densities. The approach is applied to constructing a 4-parameter Mittag-Leffler density, whose Laplace transform is a corresponding Mittag-Leffler function, which is completely monotone (CM) by construction. Laplace transforms of mixtures of the stable densities with respect to the 4-parameter Mittag-Leffler distribution are compositions of the Mittag-Leffler functions with Bernstein functions, thereby generating a rich family of CM variants of the base CM Mittag-Leffler functions, including known instances as special cases.

math.PR↗

arXiv2023

Stable Densities, Fractional Integrals and the Mittag-Leffler Function

This paper combines probability theory and fractional calculus to derive a novel integral representation of the three-parameter Mittag-Leffler function or Prabhakar function, where the three parameters are combinations of four base parameters. The fundamental concept is the Riemann-Liouville fractional integral of the one-sided stable density, conditioned on a scale factor. Integrating with respect to a gamma-distributed scale factor induces a mixture of Riemann-Liouville integrals. A particular combination of four base parameters leads to a representation of the Prabhakar function as a weighted mixture of Riemann-Liouville integrals at different scales. The Prabhakar function constructed in this manner is the Laplace transform of a four-parameter distribution. This general approach gives various known results as special cases (notably, the two-parameter generalised Mittag-Leffler distribution).

math.PR↗

arXiv2023

A Probabilistic Perspective on Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function

The main contribution of this paper is the use of probability theory to prove that the three-parameter Mittag-Leffler function is the Laplace transform of a distribution and thus completely monotone. Pollard used contour integration to prove the result in the one-parameter case. He also cited personal communication by Feller of a discovery of the result by ''methods of probability theory''. Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the result. We pursue the theme of probability theory to explore complete monotonicity beyond the contribution due to Feller. Our approach involves an interplay between mixtures and convolutions of stable and gamma densities, together with a limit theorem that leads to a novel integral representation of the three-parameter Mittag-Leffler function (also known as the Prabhakar function).

math.PR↗

arXiv2022

On Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function $E_α(-x)$

Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone. He also cited personal communication by Feller of a discovery of the result by "methods of probability theory". In his published work, Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the Pollard result. But both approaches may be described as analytic, despite the occurrence of the stable distribution in Feller's starting point and in the Pollard result itself. We adopt a Bayesian probabilistic approach that assigns a prior distribution to the scale parameter of the stable distribution. We present Feller's method as a particular instance of such assignment. The Bayesian framework enables generalisation of the Pollard result. This leads to a novel integral representation of the Mittag-Leffler function as well as a variant arising from polynomial tilting of the stable density.

math.PR↗