6 - A NOTE ON THE MATHEMATICAL THEORY OF CORRELATED RANDOM POINTS

Abstract

This chapter discusses certain aspects of the theory of singular correlated random points. Random sets of points are investigated by the correlation functions including the multiple correlations among the points. A number of expressions are derived for conditional distributions, the characteristic function of a random set of points and the Bogolyubov's deriving functional. There are cases in which one has to deal with a number of points that occupy a random position in space and such points are known as random points. Good examples of such points are the instants at which some random event occurs, the intersection of random functions with other curves and the centres of gravity of particles in statistical physics, etc. Such points are distributed in accordance with Poisson's law in that special case when there is no correlation whatever among the positions of the random points. However in the general case, the points are correlated and the formulae associated with Poisson's law are no longer valid.