Probability Schemes with Contagion in Space and Time

Abstract

In many natural assemblies of elements, the probability of an event for a given element depends not only on the intrinsic nature of that particular element, but also on the states of some or all of the rest of the elements belonging to the same assembly. On the basis of this general idea of "contagion" some urn schemes are developed in this paper in which one has contagious influence in space and time. The most interesting result found is that in general the points of convergence of the probability of the assembly are given by some of the roots of an equation $p = f(p)$ and that some of these roots, between zero and one, represent stable states of the assembly, or points of convergence, and others represent unstable ones, or points of divergence. The two neighboring roots, (if they are single), of a root representing a point of convergence are unstable values of the probability. Consequently, under certain conditions, the limiting probability may be made to have a finite jump by changing the initial probability by an arbitrarily small amount. The concrete cases developed in this paper can be considerably extended by similar methods by assuming more complicated and general assemblies and laws of contagion.