7.1 7.1 Foundations of Stochastic Point Processes
7.1.1 1.1 Poisson processes
Among stochastic point processes, most fundamental process is a Poisson process, which is characterized as follows. Let 
denote the history of the process at time t, i.e., a realization of the positions of all points in (−∞, t]. For u<v, let N(u,v) be a random variable giving the number of points in (u, v]. The Poisson process of rate μ is defined by the following requirements. For all t, as Δ t→0+,
so that
These aspects indicate that the probabilities concerned do not depend on 
. Particularly, the probability of finding a point in (t,t+Δ t] is independent of realizations of event occurrences in (−∞,t]. Furthermore, any event occurrences in (t,∞) is independent of 
. The requirement (1.2) excludes the possibility of multiple simultaneous occurrences, i.e., more than one point at the same moment, which is called ‘orderliness’. An essential property seen from (1.1) to (1.3)is that μ does not...
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Leondes, C.T. (2003). Modeling Techniques of Point Processes and Applications in Processing Biomedical Data. In: Leondes, C.T. (eds) Computational Methods in Biophysics, Biomaterials, Biotechnology and Medical Systems. Springer, Boston, MA. https://doi.org/10.1007/0-306-48329-7_7
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