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Provided that the mapping (or transformation) adheres to some conditions, then a result sometimes known as the Mapping theorem [2] says that if the original process is a Poisson point process with some intensity measure, then the resulting mapped (or transformed) collection of points also forms a Poisson point process with another intensity ...
The mapping theorem is a theorem in the theory of point processes, a sub-discipline of probability theory. It describes how a Poisson point process is altered under measurable transformations .
The simplest and most ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson process. A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution. A Poisson ...
A visual depiction of a Poisson point process starting. In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
Point Processes is a book on the mathematics of point processes, randomly located sets of points on the real line or in other geometric spaces. It was written by David Cox and Valerie Isham , and published in 1980 by Chapman & Hall in their Monographs on Applied Probability and Statistics book series.
A point process is often denoted by a single letter, [1] [7] [8] for example , and if the point process is considered as a random set, then the corresponding notation: [1], is used to denote that a random point is an element of (or belongs to) the point process . The theory of random sets can be applied to point processes owing to this ...