Fw Fred Steutel

Infinite divisibility of probability distributions on the real line

infinitely divisible distributions on the nonnegative integers infinitely divisible distributions on the nonnegative reals infinitely divisible distributions on the real line self-decomposability and stability infinite divisibility and mixtures infinite divisibility in stochastic processes. Appendices: prerequisites from probability and analysis list of well-known distributions notations and conventions.

Infinite divisibility of random variables and their integer parts

It is examined to what extent the infinite divisibility of a random variable X entails the infinite divisibility of its integer part [X] or vice versa. As a special case passage times are considered in processes with independent increments such as the positive stable processes and the Gamma process. In spite of some interesting relationships, the results tend to be rather negative.

Stability equations for processes with stationary independent increments using branching processes and Poisson mixtures

The equation X1X2W( X1+ X2)with W uniform (0,1) distributed and W,X1 and X2 independent, is generalized in several directions. Most importantly, a generalized multiplication operation is used in which subcritical branching processes, both with discrete and continuous state space, play an important role. The solutions of the equations so obtained are related to the concepts of self-decomposability and stability, both in the classical and in an extended sense. The solutions for +-valued random variables are obtained from those for +-valued random variables by way of Poisson mixtures. There are also some new results on (generalized) unimodality.

INTEGER-VALUED BRANCHING PROCESSES WITH IMMIGRATION

The notion of self-decomposability for No-valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].

Stationarity of delayed subordinators

It is shown that, analogous to partial-sum processes in renewal theory, nondecreasing Lévy processes (subordinators) can be delayed such as to show a certain stationarity.

On moment sequences and infinitely divisible sequences

We consider infinitely divisible sequences in the context of moment sequences, in the same vein as R. A. Horn [J. Math. Anal. Appl. 31 (1970), 130–135] considered renewal sequences, and we compare the results.

On multivariate infinitely divisible distributions

Simple conditions are given which characterize the generating function of a nonnegative multivariate infinitely divisible random vector. Necessary conditions on marginals, linear combinations, tail behavior, and zeroes are discussed, and a sufficient condition is given. The latter condition, which is a multivariate generalization of ordinary log-convexity, is shown to characterize only certain products of univariate infinitely divisible distributions.

Note on the approximation of distributions on Z+ by mixtures of negative binomial distributions

It is shown that the distributions on Z+ that can be approximated by mixtures of negative binomial distributions, are precisely the so-called Poisson mixtures, i.e., mixtures of Poisson distributions

Integer-valued self-similar processes

Let IN0 denote the set of nonnegative integers. We consider IN0-valued analogues of self-similar processes by defining Unvalued fractions of IN0-valued processes. These fractions are defined in terms of sums of independent Markov branching processes, in such a way that the one-dimensional marginals coincide with the IN0-valued multiples of IN0-valued random variables as introduced in [10] and [3]; this requirement still leaves room for several definitions of an 0-valued fraction, and a sensible choice has to be made. The relation with branching processes has two aspects. On the one hand, results from the theory of these processes can be used to prove analogues of classical theorems, on the other hand new results about branching processes are suggested by translating analogues of classical results in terms of branching processes. In this way, analogues are derived of the basic properties of classically self-similar processes such as obtained by Lamperti [5], and a simple relationship is established between...

Note On The Continuation Of Infinitely Divisible Distributions And Canonical Measures

We consider the question, to what extent an infinitely divisible distribution function on is determined by its values on an interval starting at zero, or by the values of its canonical measure on such an interval. These questions are considered in the book by Rossberg et al. (1985). Our results extend parts of their work. These results are applied to the continuation of constant multiples of infinitely divisible distribution functions and to the distribution of subordinators.