Point process notation

In probability and statistics, point process notation comprises the range of mathematical notation used to symbolically represent random objects known as point processes, which are used in related fields such as stochastic geometry, spatial statistics and continuum percolation theory and frequently serve as mathematical models of random phenomena, representable as points, in time, space or both. The notation varies due to the histories of certain mathematical fields and the different interpretations of point processes, and borrows notation from mathematical areas of study such as measure theory and set theory. The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain assumptions can be interpreted as random sequences of points, random sets of points or random counting measures. In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in d-dimensional Euclidean space Rd as well as some other more abstract mathematical spaces. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space Rd. A point process is called simple if no two (or more points) coincide in location with probability one. Given that often point processes are simple and the order of the points does not matter, a collection of random points can be considered as a random set of points The theory of random sets was independently developed by David Kendall and Georges Matheron. In terms of being considered as a random set, a sequence of random points is a random closed set if the sequence has no accumulation points with probability one A point process is often denoted by a single letter, for example , and if the point process is considered as a random set, then the corresponding notation: is used to denote that a random point is an element of (or belongs to) the point process .