A group of processes related to the Poisson process

Abstract

A group of stochastic processes akin to the Poisson process is defined in terms of rules of interactions between two types of interacting entities and in terms of a parameter corresponding to the initial relative numbers of the two types of interacting entities. One limiting value of this parameter corresponds to the Poisson process and the exponential distribution function, and the other limiting value of the parameter corresponds to a special case among the group of stochastic processes defined and a statistical distribution function used previously for incomes and fracture toughness. The related processes in between, which correspond to the intermediate values of the parameter, correspond to an intermediate statistical distribution function. The transition between the limiting cases is smooth as evinced by the change of the mean and median with change in the parameter. The scale-invariant behaviour of the fields of stress and strain at the tips of cracks is used to support the introduction of a shape parameter into the special-case function. All the distribution functions considered are found to be stable extreme-value functions, either in the sense of multiplying probabilities or in the sense of summing the variable or in a mixture of both senses.