Preface: graph colorings

Analogues are proposed for the concepts of self-decomposability and stability for distributions on the nonnegative integers. It turns out that these "discrete self-decomposable" and "discrete stable" distributions have properties that are quite similar to those of their continuous counterparts.

Discrete analogues of self-decomposability and stability

The article provides an historical survey of the early contri- butions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian during the period 1929-1938. In Appendix we report the translation from the Russian into English of a fundamental paper by Khintchine published in Moscow in 1937.

http://www.brunodefinetti.it/Bibliografia/Mainardi2006.pdf

The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula

We review several common discretization schemes and study a particular class of power-tail probability distributions on integers, obtained by discretizing continuous Pareto II (Lomax) distribution through one of them. Our results include expressions for the density and cumulative distribution functions, probability generating function, moments and related parameters, stability and divisibility properties, stochastic representations, and limiting distributions of random sums with discrete-Pareto number of terms. We also briefly discuss issues of simulation and estimation and extensions to multivariate setting.

Discrete Pareto Distributions

The aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Levy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.

Characterizations of discrete compound Poisson distributions

When the distribution of a random (N) sum of independent copies of a r.v X is of the same type as that of X we say that X is N-sum stable. In this paper we consider a generalization of stability of geometric sums by studying distributions that are stable under summation w.r.t Harris law. We show that the notion of stability of random sums can be extended to include the case when X is discrete. Finally we propose a method to identify the probability law of N for which X is N-sum stable.

Stability of Random Sums

A method for constructing distributions on the non-negative lattice of points Io = {0,1,2, …} as discrete analogue of continuous distributions on [0,∞ ) is presented A justification of the definition of discrete class-L laws is provided Discrete analogue of distributions of the same type and the role of Bernoulli law in this context is discussed Generalizations of some distributions and properties of α -Poisson laws are given The geometric compounding problem for discrete distributions is studied by introducing discrete semi Mittag-Leffler laws

Some classes of distributions on the non-negative lattice

Coloring the vertices of a graph G according to certain conditions is a random experiment and a discrete random variable X is defined as the number of vertices having a particular color in the given type of coloring of G and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph G is a proper coloring &#x1d49e; of G which an assignment of colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

On certain parameters of equitable coloring of graphs

In this paper, a new type of colouring called Johan colouring is introduced. This colouring concept is motivated by the newly introduced invariant called the rainbow neighbourhood number of a graph. The study ponders on maximal colouring opposed to minimum colouring. An upper bound for a connected graph is presented and a number of explicit results are presented for cycles, complete graphs, wheel graphs and for a complete $l$-partite graph.

On Certain Colouring Parameters of Graphs

Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated 01May2003) here we study certain aspects of randomization in infinitely divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and MID laws. In particular we study mixtures of ID & MID laws, its relation to random sums & random maximums, corresponding stationary processes & extremal processes and some of their properties. It is shown that mixtures of ID laws and mixtures of MID laws appear as limits of random sums and random maximums respectively. We identify a class of probability generating functions for N, the random sample size. A method to construct class-L laws is given.

S. Satheesh

Papers

Stability of Random Sums

Some classes of distributions on the non-negative lattice

On certain parameters of equitable coloring of graphs

On Certain Colouring Parameters of Graphs

Aspects of Randomization in Infinitely Divisible and Max-Infinitely Divisible Laws