On Poisson and Composed Poisson Stochastic Set Functions

Several investigations has recently been made concerning Poisson and composed Poisson stochastic processes. The ordinary Poisson process is conceivable as a sequence of points, distributed at random on the time axis and this idea can be generalized to more than onedimensional spaces. The latter case occurs in making a blood-count, in counting stars, etc. In [8], [4], [6] and [15] conditions are given ensuring the Poisson character of the distribution of the number of points in a set A of the one, resp. at least one-dimensional Euclidean space. In [8], [14], [1] and [13] similar problems are considered for the onedimensional Euclidean space and the main purpose is to prove that under some conditions the random variables ξt2 − ξt1 (t1 < t2) have composed Poisson distributions. We say that a random variable ξ has a composed Poisson distribution if its characteristic function f(u) can be written in the form