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Zbigniew J. Jurek

Publications and source records attributed to Zbigniew J. Jurek.

35 records · Page 2Linked to original sources
arXiv2013

Remarks on factorization property of some stochastic integrals

In the paper Sato (2006) there are introduced two families of improper random integrals and the corresponding two convolution semigroups of infinitely divisible laws on $\Rset^d$. Theorem 3.1 gives a relation (a factorization property) between those two integrals. Here, using \emph{the random integral mappings} $I^{h,r}_{(a,b]}$ (cf. the survey article Jurek (2011)), we give a simpler proof that is also valid for measures on Banach spaces. Furthermore, using our technique we establish yet other relations between those two families of improper stochastic integrals.

math.PR↗
arXiv2013

Selfdecomposability, perpetuity and stopping times

In the probability theory limit distributions (or probability measures) are often characterized by some convolution equations (factorization properties) rather than by Fourier transforms (the characteristic functionals). In fact, usually the later follows the first one. Equations, in question, involve the multiplication by the positive scalars $c$ or an action of the corresponding dilation $T_{c}$ on measures. In such a setting, it seems that there is no way for stopping times (or in general, for the stochastic analysis) to come into the "picture". However, if one accepts the view that the primary objective, in the classical limit distributions theory, is to describe the limiting distributions (or random variables) by the tools of random integrals/functionals then one can use the stopping times. In this paper we illustrate such a possibility in the case of selfdecomposability random variables (i.e. Lévy class L) with values in a real separable Banach space. Also some applications of our approach to perpetuity laws are presented; cf. \cite{Du1}, \cite{ED1}, \cite{GG1}. In fact, we show that all selfdecomposable distributions are perpetuity laws.

math.PR↗
arXiv2013

A random integral calculus on generalized s-selfdecomposable probability measures

It is known that the class $\mathcal{U}_β$, of generalized s-selfdecom-posable probability distributions, can be viewed as an image via random integral mapping $\mathcal{J}^β$ of the class $ID$ of all infinitely divisible measures. We prove that a composition of the mappings $\mathcal{J}^{β_1}, \mathcal{J}^{β_2}, ..., \mathcal{J}^{β_n}$ is again random integral mapping but with a new inner time. In a proof some form of Lagrange interpolation formula is needed. Moreover, some elementary formulas concerning the distributions of products of powers of independent uniformly distributed random variables as established as well.

math.PR↗
arXiv2012

Random integral representations for free-infiniteley divisible and tempered distributions

There are given sufficient conditions under which mixtures of dilations of Lévy spectral measures, on a Hilbert space, are Lévy measures again. We introduce some random integrals with respect to infinite dimensional Lévy processes, which in turn give some integral mappings. New classes (convolution semigroups) are introduced. One of them gives an unexpected relation between the free (Voiculescu) and the classical Lévy-Khintchine formulae while the second one coincides with tempered stable measures (Mantegna nad Stanley) arisen in statistical physics.}

math.PR↗
arXiv2012

Invariant measures under random integral mappings and marginal distributions of fractional Lévy processes

It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of generalized s-selfdecomposable and selfdecomposable distributions. Some application are given to the moving average fractional Lévy process (MAFLP).

math.PR↗
arXiv2012

Remarks on restricted Nevanlinna transforms

The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted Nevanlinna transform) to the imaginary axis can be expressed as the Laplace transform of the Fourier transform (characteristic function) of the corresponding measure. Finally, a relation between the Voiculescu and the boolean convolution is indicated.

math.PR↗
arXiv2012

The Random Integral Representation Conjecture: a quarter of a century later

In Jurek 1985 and 1988 the random integral representations conjecture was stated. It claims that (some) limit laws can be written as probability distributions of random integrals of the form $\int_{(a,b]}h(t)dY_ν(r(t))$, for some deterministic functions $h$, $r$ and a Lévy process $Y_ν(t),t\ge 0$. Here we review situations where a such claim holds true. Each theorem is followed by a remark which gives references to other related papers, results as well as some historical comments. Moreover, some open questions are stated.

math.PR↗
arXiv2010

Selfdecomposable Laws Associated with Hyperbolic Functins

It is shown that the hyperbolic functions can be associated with selfdecomposable distributions (in short: SD probability distributions or Lévy class L probability laws). Consequently, they admit associated background driving Lévy processes $Y$ (BDLP Y). We interpret the distributions of Y(1) via Bessel squared processes, Bessel bridges and local times.

math.PR↗
arXiv2010

Generalized L/'evy Stochastic Areas and Selfdecomposability

We show that a conditional characteristic function of generalized Lévy stochastic areas can be viewed as a product a selfdecomposable distribution (i.e., Lévy class L distribution) and its background driving characteristic function. This provides a stochastic interpretation for a ratio of some Bessel functions as well as examples of characteristic functions from van Dantzig class.

math.PR↗
arXiv2008

Factorization property of generalized s-selfdecomposable measures and class $L^f$ distributions$^1$

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↗
arXiv2008

A note on a composition of two random integral mappings $\J^\be$ and some examples

A method of random integral representation, that is, a 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 show that a composition of two random integral mappings $\J^\be$ is again a random integral mapping. We illustrate our results on some examples.

math.PR↗
arXiv2008

A calculus on Lévy exponents and selfdecomposability on Banach spaces

In infinite dimensional Banach spaces there is no complete characterization of the Lévy exponents of infinitely divisible probability measures. Here we propose \emph{a calculus on Lévy exponents} that is derived from some random integrals. As a consequence we prove that \emph{each} selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov-Jurek-Schreiber in the Annals of Probability \textbf{32}, 2004.}

math.PR↗
arXiv2008

On Relations Between Urbanik and Mehler Semigroups

It is shown that operator-selfdecomposable measures, or more precisely their Urbanik decomposability semigroups, induce generalized Mehler semigroups of bounded linear operators. Moreover, those semigroups can be represented as random integrals of operator valued functions with respect to stochastic Lévy processes. Our Banach space setting is in the contrast with the Hilbert spaces on which so far and most often the generalized Mehler semigroups were studied. Furthermore, we give new proofs of the random integral representation.

math.PR↗
arXiv2005

A new factorization property of the selfdecomposable probability measures

We prove that the convolution of a selfdecomposable distribution with its background driving law is again selfdecomposable if and only if the background driving law is s-selfdecomposable. We will refer to this as the factorization property of a selfdecomposable distribution; let L^f denote the set of all these distributions. The algebraic structure and various characterizations of L^f are studied. Some examples are discussed, the most interesting one being given by the Levy stochastic area integral. A nested family of subclasses L^f_n, n\ge 0, (or a filtration) of the class L^f is given.

math.PR↗
arXiv2003

On fixed points of Poisson shot noise transforms

Distributional fixed points of a Poisson shot noise transform (for nonnegative, nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than -1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

math.PR↗
arXiv2002

Shot noise distributions and selfdecomposability

Stationary (limiting) distributions of shot noise processes, with exponential response functions, form a large subclass of positive selfdecomposable distributions that we illustrate by many examples. These shot noise distributions are described among selfdecomposable ones via the regular variation at zero of their distribution functions. However, slow variation at the origin of (an absolutely continuous) distribution function is incompatible with selfdecomposability and this is shown in three examples.

math.PR↗