Tetyana Kadankova

Bio: Tetyana Kadankova is an academic researcher from Vrije Universiteit Brussel. The author has contributed to research in topics: Bessel function & Poisson distribution. The author has an hindex of 1, co-authored 3 publications receiving 6 citations.

Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

Abstract: We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Polya-Aeppli process of order k: We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

Two further probabilistic applications of Bessel functions

Abstract: We revisit two classical formulas for the Bessel function of the first kind, due to von Lommel and Weber-Schafheitlin, in a probabilistic setting. The von Lommel formula exhibits a family of solutions to the van Dantzig problem involving the generalized semi-circular distributions and the first hitting times of a Bessel process with positive parameter, whereas the Weber-Schafheitlin formula allows one to construct non-trivial moments of Gamma type having a signed spectral measure. Along the way, we observe that the Weber-Schafheitlin formula is a simple consequence of the von Lommel formula, the Fresnel integral and the Selberg integral.

Remark on a Mittag–Leffler function of Le Roy type

Abstract: We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag–Leffler function of Le Roy type. It is conjectured that the underlying posi...

Generalized Fractional Counting Process

Abstract: In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. [8]. For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied, using which its long-range dependence property is established. It is shown that the increments of GFCP exhibit the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP), is discussed for which we obtain a limiting result and a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order k, the Pólya-Aeppli process of order k, the negative binomial process and their fractional versions etc., are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

On multivariate quasi-infinitely divisible distributions

Abstract: A quasi-infinitely divisible distribution on $\mathbb{R}^d$ is a probability distribution $\mu$ on $\mathbb{R}^d$ whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on $\mathbb{R}^d$. Equivalently, it can be characterised as a probability distribution whose characteristic function has a L\'evy--Khintchine type representation with a "signed L\'evy measure", a so called quasi--L\'evy measure, rather than a L\'evy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato \cite{lindner}. The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on $\mathbb{Z}^d$-valued quasi-infinitely divisible distributions.

Generalized Fractional Counting Process

Abstract: In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. (2016). For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied using which its long-range dependence property is established. It is shown that the increments of GFCP exhibits the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP) is discussed for which we obtain a limiting result, a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order $k$, the Polya-Aeppli process of order $k$, the negative binomial process and their fractional versions etc. are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

Remark on a Mittag-Leffler function of Le Roy type

Abstract: We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag-Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible.