Point process operation

In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space. These operations can be purely random, deterministic or both, and are used to construct new point processes, which can be then also used as mathematical models. The operations may include removing or thinning points from a point process, combining or superimposing multiple point processes into one point process or transforming the underlying space of the point process into another space. Point process operations and the resulting point processes are used in the theory of point processes and related fields such as stochastic geometry and spatial statistics. One point process that gives particularly convenient results under random point process operations is the Poisson point process, The Poisson point process often exhibits a type of mathematical closure such that when a point process operation is applied to some Poisson point process, then provided some conditions on the point process operation, the resulting process will be often another Poisson point process operation, hence it is often used as a mathematical model. Point process operations have been studied in the mathematical limit as the number of random point process operations applied approaches infinity. This had led to convergence theorems of point process operations, which have their origins in the pioneering work of Conny Palm in 1940s and later Aleksandr Khinchin in the 1950s and 1960s who both studied point processes on the real line, in the context of studying the arrival of phone calls and queueing theory in general. Provided that the original point process and the point process operation meet certain mathematical conditions, then as point process operations are applied to the process, then often the resulting point process will behave stochastically more like a Poisson point process if it has a non-random mean measure, which gives the average number of points of the point process located in some region.