arXiv subjects

Zbigniew J. Jurek

Publications and source records attributed to Zbigniew J. Jurek.

At least 19 recordsLinked to original sources

arXiv2023

Some definite integrals arising from selfdecomposable characteristic functions

In the probability theory \emph{selfdecomposable, or class $L_0$ distributions} play an important role as they are limiting distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class $L_0$ is quite large and includes many known classical distributions and statistics. For this note the most important feature of the selfdecomposable variables are their random integral representation with respect to Lévy process. From those random integral representation we get equality of logarithms of some characteristic functions. These allows us to get formulas for some definite integrals, some of them probably were unknown before.

math.PR↗

arXiv2022

Which Urbanik class $L_k$, do the hyperbolic and the generalized logistic characteristic functions belong to?

Selfdecomposable variables obtained from series of Laplace (double exponential) variables are objects of this study. We proved that hyperbolic-sine and hyperbolic-cosine variables are in the difference of the Urbanik classes $L_2$ and $L_3$ while generalized logistic variable is at least in the Urbanik class $L_1$. Hence some ratios of those corresponding selfdecomposable characteristic functions are again selfdecomposable.

math.PR↗

arXiv2022

Background driving distribution functions and series representation for log-gamma selfdecomposable random variables

For the selfdecomposable distributions (random variables) we identified background driving probability distributions in their random integral representations. For log-gamma and their background driving random variables series representations are found.

math.PR↗

arXiv2022

Urbanik type subclasses of the free-infinitely divisible transforms

For the class of free-infinitely divisible transforms are introduced three families of increasing Urbanik type subclasses of those transforms. They begin with the class of free-normal transforms and end up with the whole class of free-infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which are known their random integral representations. Special functions like Hurwitz-Lerch, polygamma and hypergeometric appeared in kernels of the corresponding integral representations.

math.PR↗

arXiv2022

On background driving distribution functions (BDDF) for some selfdecomposable variables

Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via Lévy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.

math.PR↗

arXiv2021

Some characterizations of multiple selfdecomposability with extensions and an application to the Gamma function

Inspirations for this paper can be traced to Urbanik (1972) where convolution semigroups of multiple decomposable distributions were introduced. In particular, the classical gamma $\mathbb{G}_t$ and $\log \mathbb{G}_t$, $t>0$ variables are selfdecomposable. In fact, we show that $\log \mathbb{G}_t$ is twice selfdecomposable if, and only if, $t\geq t_1 \approx 0.15165$. Moreover, we provide several new factorizations of the Gamma function and the Gamma distributions. To this end, we revisit the class of multiply selfdecomposable distributions, denoted $L_n(R)$, and propose handy tools for its characterization, mainly based on the Mellin-Euler's differential operator. Furthermore, we also give a perspective of generalization of the class $L_n(R)$ based on linear operators or on stochastic integral representations.

math.PR↗

arXiv2021

Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view

We establish a new connection between the class of Nevanlinna-Pick functions and the one of the exponents associated to spectrally negative Lévy processes. As a consequence, we compute the characteristics related to some hyperbolic functions and we show a property of temporal complete monotonicity, similar to the one obtained via the Lamperti transformation by Bertoin \& Yor ({\it On subordinators, self-similar Markov processes and some factorizations of the exponential variable}, Elect. Comm. in Probab., vol. 6, pp. 95--106, 2001) for self-similar Markov processes. More precisely, we show the remarkable fact that for a subordinator $ξ$, the function $t \mapsto t^n \, \er[ξ_t^{-p}]$ is , depending on the values of the exponents $n=0,1,2,\; p>-1$, or a Bernstein function or a completely monotone function. In particular, $ξ$ is the inverse time subordinator of a spectrally negative Lévy process, if, and only if, for some $\,p\geq 1$, the function $t \mapsto t \, \er[ξ_t^{-p}]$ is a Stieltjes transform. Finally, we clarify to which extent Nevanlinna-Pick functions are related to free-probability and to Voiculescu transforms, and we provide an inversion procedure.

math.PR↗

arXiv2021

On a limit theorem for a non-linear scaling

In this note, we proved that weak limits, of sums of independent positive identically distributed random variables which are re-normalized by a non-linear shrinking transform $\max(0, x-r)$, are either degenerate or (some) compound Poisson distributions.

math.PR↗

arXiv2021

Remarks on the tangent function from an analytic and probability point of view

We show that a function $\tan(1/it)$ is a Pick function (free-infinitely divisible transform) and indicate its connections with a probability. Moreover, we found its "counterpart" in classical infinitely divisible measures expressed as series of Rademacher variables.

math.PR↗

arXiv2020

Remarks on compositions of some random integral mappings

The random integral mappings (some type of functionals of Lévy processes) are continuous homomorphisms between convolution subsemigroups of the semigroup of all infinitely divisible measures. Compositions of those random integrals (mappings) can be always expressed as another single random integral mapping. That fact is illustrated by some old and new examples.

math.PR↗

arXiv2020

On hyperbolic characteristic functions from an analytic and a free-probability point of view

For free-probability Voiculescu transforms, analogous to hyperbolic characteristic functions, we show how to get their representing measures in an integral form. For that purpose, it is enough to know those transforms only on the imaginary axis. This is in contrast to a complex analysis where one needs to know them in some domains in the complex plane.

math.PR↗

arXiv2017

Ona relation between classical and free infinitely divisible transforms

We study two ways (levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms. For free-selfdecomposable measures we found the formula (a differential equation) for their background driving transforms. We illustrate our methods on the hyperbolic characteristic functions. As a by-product our approach may produce new formulas for some definite integrals.

math.PR↗

arXiv2014

On a method of introducing free-infinitely divisible probability measures

Random integral mappings $I^{h,r}_{(a,b]}$ give isomorphisms between the sub-semigroups of the classical $(ID, \ast)$ and the free-infinite divisible $(ID,\boxplus)$ probability measures. This allows us to introduce new examples of such measures and their corresponding characteristic functionals.

math.PR↗

arXiv2014

On a central limit theorem for shrunken weakly dependent random variables

A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For independent, identically distributed random variables, this result was proved earlier by Housworth and Shao.

math.PR↗

arXiv2014

The strong mixing and the operator-selfdecomposability properties

For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the possible limit distributions are precisely the operator-selfdecomposable laws.

math.PR↗

arXiv2014

Factorization Property of Generalized s-self-decomposable measures and class $L^f$ distributions

The method of \emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \emph{factorization property}. These are the classes $\mathcal{U}^f_{\be}$ and $L^f$, respectively.

math.PR↗

arXiv2013

Calculus on random integral mappings $I^{h,r}_{(a,b]}$ and their domains

It is proved that the random integral mappings (some type of functionals of Lévy processes) are always isomorphisms between convolution semigroups of infinitely divisible measures. However, the inverse mappings are no longer of the random integral form. Domains are characterized in may ways. Compositions (iterated integrals) can be expressed as a single random integral mapping. Finally, all obtained results are illustrated by examples.

math.PR↗

arXiv2013

The random integral representation hypothesis revisited : new classes of s-selfdecomposable laws

For $\,0<α\le \infty$, new subclasses $\,\mathcal{U}^{<α>}$ of the class $\,\mathcal{U}$, of s-selfdecomposable probability measures, are studied. They are described by random integrals, by their characteristic functions and their Lévy spectral measures. Also their relations with the classical Lévy class $L$ of selfdecomposable distributions are investigated.

math.PR↗