Tomasz J. Kozubowski

A Multivariate and Asymmetric Generalization of Laplace Distribution

SummaryConsider a sum of independent and identically distributed random vectors with finite second moments, where the number of terms has a geometric distribution independent of the summands. We show that the class of limiting distributions of such random sums, as the number of terms converges to infinity, consists of multivariate asymmetric distributions that are natural generalizations of univariate Laplace laws. We call these limits multivariate asymmetric Laplace laws. We give an explicit form of their multidimensional densities and show representations that effectively facilitate computer simulation of variates from this class. We also discuss the relation to other formerly considered classes of distributions containing Laplace laws.

Exponential Mixture Representation of Geometric Stable Distributions

We show that every strictly geometric stable (GS) random variable can be represented as a product of an exponentially distributed random variable and an independent random variable with an explicit density and distribution function. An immediate application of the representation is a straightforward simulation method of GS random variables. Our result generalizes previous representations for the special cases of Mittag-Leffler and symmetric Linnik distributions.

Mixture representation of Linnik distribution revisited

Let Y[alpha] have a Linnik distribution, given by the characteristic function [psi](t) = (1 + t [alpha])-1. We extend the result of Kotz and Ostrovskii (1996) and show that Y[alpha] admits two different representations, where 0

Representation and Properties of Geometric Stable Laws

A random summation scheme, where the number of terms is geometrically distributed, is called a geometric summation scheme (geometric compound, geometric convolution) (Klebanow et al., 1984). Geometric convolutions naturally arise in many applied probability problems. In particular, they appear in queueing theory and reliability in connection to“regenerating processes with rare events”(Gertsbakh, 1984; Jacobs, 1986). Some recent results suggest that geometric compounds could provide useful models in economics (Kozubowski and Rachev, 1992).

On moments and tail behavior of v-stable random variables

In this paper a class of limiting probability distributions of normalized sums of a random number of i.i.d. random variables is considered. The representation of such distributions via stable laws and asymptotic behavior of their moments and tail probabilities are established.

Computer simulation of geometric stable distributions

We present a new method for computer simulation of strictly geometric stable random variables. The method is based on their representation as a product of two independent random variables with explicit distribution functions, coupled with the inversion method. We also extend the method to the multivariate case, by deriving new representation and simulation algorithm for multivariate Linnik distribution.