# Fluctuations of extremal Markov chains driven by the Kendall convolution

@inproceedings{JasiulisGodyn2019FluctuationsOE, title={Fluctuations of extremal Markov chains driven by the Kendall convolution}, author={Barbara H. Jasiulis-Gołdyn and Edward Omey and Mateusz Staniak}, year={2019} }

. The paper deals with ﬂuctuations of Kendall random walks, which are extremal Markov chains and iterated integral transforms with the Williamson kernel Ψ( t ) = (1 − | t | α ) + , α > 0. We obtain the joint distribution of the ﬁrst ascending ladder epoch and height over any level a 0 and distribution of maximum and minimum for these extremal Markovian sequences solving recursive integral equations. We show that distribution of the ﬁrst crossing time of level a 0 is a mixture of geometric and…

## References

SHOWING 1-10 OF 37 REFERENCES

### Kendall random walks

- Mathematics
- 2014

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…

### Introductory Lectures on Fluctuations of Lévy Processes with Applications

- Mathematics
- 2006

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance…

### An extremal markovian sequence

- MathematicsJournal of Applied Probability
- 1989

In this paper we consider an independent and identically distributed sequence {Yn } with common distribution function F(x) and a random variable X 0, independent of the Yi 's, and define a Markovian…

### The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution

- Mathematics
- 2012

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…

### Kendall random walk,Williamson transform, and the corresponding Wiener–Hopf factorization

- Mathematics
- 2015

We give some properties of hitting times and an analogue of the Wiener–Hopf factorization for the Kendall random walk. We also show that the Williamson transform is the best tool for problems…

### Classical definitions of the Poisson process do not coincide in the case of generalized convolutions

- Mathematics
- 2013

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,…

### L\'evy processes with respect to the index Whittaker convolution

- Mathematics
- 2018

The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a…

### Lévy processes and infinitely divisible distributions

- Mathematics
- 2013

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5.…

### Limit Property for Regular and Weak Generalized Convolution

- Mathematics
- 2010

AbstractWe denote by ℘
$(\mathcal{P_{+}})$
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…