# Barbara HELENA Jasiulis-GołdynUniversity of Wroclaw | WROC · Department of Mathematics and Computer Science

Barbara HELENA Jasiulis-Gołdyn

PhD. Mathematics MA Painting

## About

21

Publications

1,302

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105

Citations

Citations since 2016

## Publications

Publications (21)

Smog is a serious problem in most big urban areas. We rarely realize the consequences of being in polluted air, while mixture of air pollutants can seriously endanger human health. Bronchitis, pneumonia and asthma are only some of the respiratory diseases that are associated with the effects of smog. Polluted air also makes it difficult for people...

Kingman, in his seminal work [13], introduced a new type of convolution of distributions that is naturally related to spherically symmetric random walks. Motivated by this paper, Urbanik in a series of papers [17] established a theory of generalized convolutions ⋄ as certain binary commutative and associative operations that include classical and K...

The World Health Organization considers pollen mixtures PM2.5 to be the most harmful to health among other types of atmospheric pollution. Particles with a smaller diameter more easily enter the body. The first contact of these dusts with the human body occurs in the respiratory tract. Our main goal is to analyze the impact of air pollution indicat...

An elementary renewal theorem and a Blackwell theorem provided by Jasiulis-Go{\l}dyn et al. (2020) in a setting of Kendall convolutions are proved under weaker hypothesis and extended to the Gamma class. Convergence rates of the limits concerned in these theorems are analyzed.

In the paper we describe several important properties of the Kendall convolution at the same time pointing to these generalized convolutions which have the same property. For example the monotonic property is necessary to build a renewal process with respect to generalized convolution, lack of memory property is needed for the construction of the P...

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions...

The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains. We give the joint distribution of the first ascending ladder epoch and height over any level $a \geq 0$ and distribution of maximum and minimum for these extremal Markovian sequences. We show that distribution of the first crossing time of level $a \geq0$ i...

We consider a class of max-AR(1) sequences connected with the Kendall convolution. For a large class of step size distributions we prove that the one dimensional distributions of the Kendall random walk with any unit step distribution, are regularly varying. The finite dimensional distributions for Kendall convolutions are given. We prove convergen...

We consider here the Cramer-Lundberg model based on generalized convolutions. In our model the insurance company invests at least part of its money, have employees, shareholders. The financial situation of the company after paying claims can be even better than before. We compute the ruin probability for $\alpha$-convolution case, maximal convoluti...

The paper deals with the renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes connected with generalized convolutions. We prove an analogue of the Fredholm theorem for all generalized convolutions algebras. Using the technique of regularly varyin...

The paper gives some properties of hitting times and an analogue of the
Wiener-Hopf factorization for the Kendall random walk. We show also that the
Williamson transform is the best tool for problems connected with the Kendall
generalized convolution.

The paper deals with a new class of random walks strictly connected with the
Pareto distribution. We consider stochastic processes in the sense of
generalized convolution or weak generalized convolution following the idea
given in [1]. The processes are Markov processes in the usual sense. Their
structure is similar to perpetuity or autoregressive...

A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$
there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}'
\stackrel{d}{=} {\bf X} \Theta$, where $X'$ is an independent copy of $X$ and
$\Theta$ is independent of $X$. This is equivalent (see [12]) with the
condition that for all random variables $Q_1, Q_2$ t...

In the paper, we consider a generalization of the notion of Poisson process to the case where the classical convolution is replaced by the generalized convolution in the sense of Urbanik [K. Urbanik, Generalized convolutions, Stud. Math., 23:217–245, 1963] following two classical definitions of the Poisson process. First, for every generalized conv...

In this paper, we present a comprehensive theory of generalized and weak
generalized convolutions, illustrate it by a large number of examples, and
discuss the related infinitely divisible distributions. We consider L\'evy and
additive process with respect to generalized and weak generalized convolutions
as certain Markov processes, and then study...

The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probabi...

Kendall (Foundations of a theory of random sets, in Harding, E.F., Kendall, D.G. (eds.), pp.322–376, Willey, New York, 1974) showed that the operation
à1\colon P+2® P+\diamond_{1}\colon \mathcal{P}_{+}^{2}\rightarrow \mathcal{P}_{+} given by
dxà1d1=xp2+(1-x)d1,\delta_x\diamond_1\delta_1=x\pi_2+(1-x)\delta_1,
where x∈[0,1] and π
β
is the Paret...

We denote by ℘
(P+)(\mathcal{P_{+}})
the set of all probability measures defined on the Borel subsets of the real line (the positive half-line[0,∞)). K.Urbanik
defined the generalized convolution as a commutative and associative ℘+-valued binary operation • on ℘+2 which is continuous in each variable separately. This convolution is distributive w...

A random vector is weakly stable iff for all there exists a random variable [Theta] such that . This is equivalent (see Misiewicz etÂ al.Â [Misiewicz, J.K., Oleszkiewicz, K., Urbanik, K., 2005. Classes of measures closed under mixing and convolution. Weak stability. Stud. Math. 167 (3), 195-213]) to the condition that for all random variables Q1,Q2...

## Projects

Project (1)

We consider max-AR(1) sequences of the Kendall type, because the distributions associated with them are heavy tailed and we apply them to air pollution modeling.
The main goals of the project:
-> Cramer-Lundberg model with applications of Kendall random walk;
-> Asymptotic properies of extremal Markovian sequences of the Kendall type;
-> Renewal theory for Kendall random walks ;
-> Wiener-Hopf factorization for max-AR(1) sequences of the Kendall type.