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1
Content available remote Urbanik type subclasses of free-infinitely~divisible~transforms
Jurek Zbigniew J.
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 1
23--39
EN
For the class of free-infinitely divisible transforms we introduce three families of increasing Urbanik type subclasses. 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 random integral repre- sentations are known. Special functions like Hurwitz–Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.
2
Content available remote On a Relation Between Classical and Free Infinitely Divisible Transforms
Jurek Zbigniew J.
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
349--367
EN
We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.
3
Content available remote Series representation of time-stable stochastic processes
Kopp C., Molchanov I.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 2
299--315
EN
A stochastically continuous process ξ(t), t ≥ 0, is said to be time-stable if the sum of n i.i.d. copies of ξ equals in distribution the time-scaled stochastic process ξ(nt), t ≥ 0. The paper advances the understanding of time-stable processes by means of their LePage series representations as the sum of i.i.d. processes with the arguments scaled by the sequence of successive points of the unit intensity Poisson process on [0,∞). These series yield numerous examples of stochastic processes that share one-dimensional distributions with a Lévy process.
4
Content available remote Fractional negative binomial and Pólya processes
Vellaisamy P., Maheshwari A.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 1
77--101
EN
In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process (SFPP) is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.
5
Content available remote On Infinite Divisibility of Convolution and Mapping Kernels
Shin K.
Fundamenta Informaticae
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2017
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Vol. 152, nr 1
87--105
EN
Determining whether convolution and mapping kernels are always infinitely divisible has been an unsolved problem. The mapping kernel is an important class of kernels and is a generalization of the well-known convolution kernel. The mapping kernel has a wide range of application. In fact, most of kernels known in the literature for discrete data such as strings, trees and graphs are mapping (convolution) kernels including the q-gram and the all-sub-sequence kernels for strings and the parse-tree and elastic kernels for trees. On the other hand, infinite divisibility is a desirable property of a kernel, which claims that the c-th power of the kernel is positive definite for arbitrary c ∈ (0, ∞). This property is useful in practice, because the c-th power of the kernel may have better power of classification when c is appropriately small. This paper shows that there are infinitely many positive definite mapping kernels that are not infinitely divisible. As a corollary to this discovery, the q-gram, all-sub-sequence, parse-tree or elastic kernel turns out not to be infinitely divisible. Although these are a negative result, we also show a method to approximate the c-th power of a kernel with a positive definite kernel under certain conditions.
6
Content available remote Convergence of the fourth moment and infinite divisibility
Arizmendi O.
Probability and Mathematical Statistics
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2013
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Vol. 33, Fasc. 2
201--212
EN
In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly (2012).
7
Content available remote A functional equation that leads to semistability
Bouzar N.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
87--102
EN
Some functional equations related to the notion of semistability of probability distributions on Z+ and R+ are studied. The solution sets of these equations are fully described.
8
Content available remote Distributional properties of the negative binomial Lévy process
Kozubowski T. J., Podgórski K.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 1
43--71
EN
The geometric distribution leads to a Lévy process parameterized by the probability of success. The resulting negative binomial process (NBP) is a purely jump and non-decreasing process with general negative binomial marginal distributions. We review various stochastic mechanisms leading to this process, and study its distributional structure. These results enable us to establish strong convergence of the NBP in the supremum norm to the gamma process, and lead to a straightforward algorithm for simulating sample paths. We also include a brief discussion of estimation of the NPB parameters, and present an example from hydrology illustrating possible applications of this model.
9
Content available remote On the remarkable distributions of maxima of some fragments of the standard reflecting random walk and Brownian motion
Fujita T., Yor M.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 1
89--104
EN
In this paper, we consider some distributions of maxima of excursions and related variables for standard random walk and Brownian motion. We discuss the infinite divisibility properties of these distributions and calculate their Lévy measures. Lastly we discuss Chung's remark related with Riemann's zeta functional equation.
10
Content available remote Lévy processes and self-decomposability in finance
Bingham N. H.
Probability and Mathematical Statistics
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2006
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Vol. 26, Fasc. 2
367--378
EN
The main theme of Urbanik’s work was infinite divisibility and its ramifications. The aim of this memorial article is to trace the application of this theme in mathematical finance, one of the main growth areas in contemporary probability theory. We begin in Section 1 with a discussion of the nature of prices. In particular, we focus on whether (or when) prices may be taken as continuous, with a view to using Lévy processes to model the case of prices with jumps. We turn in Section 2 to asset return distributions; prime candidates for modelling here include the normal, hyperbolic and Student t cases. In Section 3, we turn to distributions of type G, in particular, those in which the mixing law is not only infinitely divisible but also self-decomposable (i.e. in the class SD), which includes all three cases above. Then in Section 4 we turn to the dynamic counterpart of this, in which the law of class SD occurs as the limit law of a stochastic process of Ornstein-Uhlenbeck type, with Lévy driving noise. Finally, in Section 5 we discuss stochastic volatility models.
11
Content available remote The class of type G distributions of Rd and related subclasses of infinitely divisible distribution
Maejima M., Rosiński J.
Demonstratio Mathematica
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2001
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Vol. 34, nr 2
251-266
EN
Classes of infinitely divisible distributions obtained by iteration of Gaus-sian randomization of Levy measures are introduced and studied. Their relation to Urbanik-Sato nested classes of selfdecomposable distributions is also established.
12
Content available remote Remarks on the selfdecomposability and new examples
Jurek Z.
Demonstratio Mathematica
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2001
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Vol. 34, nr 2
241-250
EN
The analytic property of the sel fdecomposability of characteristic functions is presented from stochastic processes point of view. This provides new examples or proofs, as well as a link between the stochastic analysis and the theory of characteristic functions. A new interpretation of the famous Levy's stochastic area formula is given.
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