# Stochastic point process

*point process*

A stochastic process corresponding to a sequence of random variables , , on the real line . Each value corresponds to a random variable called its multiplicity. In queueing theory a stochastic point process is generated by the moments of arrivals for service, in biology by the moments of impulses in nerve fibres, etc.

The number of all points is called the counting process, , where is a martingale and is the compensator with respect to the -fields generated by the random points . Many important problems can be solved in terms of properties of the compensator .

Let be a complete separable metric space, the class of bounded Borel sets , the set of all measures that take integral values, , and the minimal -field generated by the subsets of measures for and . Specifying a probability measure in the measurable space determines a stochastic point process with state space whose realizations are integer-valued measures on . The values for which are called the points of . The quantity is equal to the sum of the multiplicities of the points of that lie in . is called simple if for all and ordinary if, for all and , there is a partition of such that

Ordinary stochastic point processes are simple. An important role is played by the factorial moment measures

and their extensions ( is the mathematical expectation and is called the measure of intensity). If , then

A special role in the theory of stochastic point processes is played by Poisson stochastic point processes , for which: a) the values of on disjoint are mutually-independent random variables (the property of absence of after-effect); and b)

For a simple stochastic point process,

(*) |

where the infimum is taken over all partitions of . The relation (*) makes it possible to find explicit expressions for the measure of intensity for many classes of stochastic point processes generated by stochastic processes or random fields.

A generalization of stochastic point processes are the so-called marked stochastic point processes, in which marks from some measurable space are assigned to points with . The service times in a queueing system can be regarded as marks.

In the theory of stochastic point processes, an important role is played by relations connecting, in a special way, given conditional probabilities of distinct events (Palm probabilities). Limit theorems have been obtained for superposition (summation), thinning out and other operations on sequences of stochastic point processes. Various generalizations of Poisson stochastic point processes are widely used in applications.

#### References

[1] | A.Ya. Khinchin, "Mathematical methods in the theory of queueing" , Griffin (1960) (Translated from Russian) |

[2] | D.R. Cox, V. Isham, "Point processes" , Chapman & Hall (1980) |

[3] | J. Kerstan, K. Matthes, J. Mecke, "Infinitely divisible point processes" , Wiley (1978) (Translated from German) |

[4] | Yu.K. Belyaev, "Elements of the general theory of point processes" (Appendix to Russian translation of: H. Cramér, M. Leadbetter, Stationary and related stochastic processes, Wiley, 1967) |

[5] | R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , II. Applications , Springer (1978) (Translated from Russian) |

[6] | M. Jacobson, "Statistical analysis of counting processes" , Lect. notes in statistics , 12 , Springer (1982) |

#### Comments

Let , be as above; let be the Borel field of . Let be the collection of all Borel measures on . For each , defines a mapping , and is the -field generated by those mappings, i.e. the smallest -field making all these mappings measurable. The integral-valued elements of form the subspace and is the induced -field on .

A random measure on is simply a probability measure on or, equivalently, a measurable mapping of some abstract probability space into . A point process is the special case that takes its values in .

An element is simple if or for all . A simple point process is one that takes its values in the subspace of consisting of the simple measures.

Each defines a function , , and, hence, gives a random measure , a random variable which will be denoted by . One can think of a random measure in two ways: a collection of measures (on ) parametrized by a probability space or a collection of random variables (on or on ) indexed by , depending on which part of the mapping one focuses on.

More generally, for each bounded continuous function on one has the random variable defined by

For each random measure one defines the Palm distributions of . For a simple point process the Palm distribution can be thought of as the conditional distribution of given that has an atom at . Palm distributions are of great importance in random measure theory and have applications to queueing theory, branching processes, regenerative sets, stochastic geometry, statistical mechanics, and insurance mathematics (the last, via doubly stochastic Poisson processes, also called Cox processes, which are Poisson processes with stochastic variation in the intensity).

The Palm distribution of a random measure is obtained by disintegrating its Campbell measure on , which is given by

for , , where is the indicator function of , the function is the (pointwise) product of the two function and and stands for expectation.

Disintegration of a measure is much related to conditional distributions (cf. Conditional distribution). Given two measurable spaces and , a kernel, also called a Markov kernel, from to is a mapping such that is measurable on for all and such that is a -finite measure on for all .

Given a -finite measure on the product space , a disintegration of consists of a -finite measure on and a kernel from to such that -almost everywhere and such that for all ,

(a1) |

It follows that for every measurable function ,

(a2) |

The inverse operation is called mixing. Given and , the measure (a1) is called the mixture of the with respect to (and (a2) could be called the Fubini formula for mixture measures).

A disintegration exists for a -finite if is Polish Borel. This reduces to a matter of conditional distributions. The measure is unique up to equivalence, and is unique up to a measurable renormalization -almost everywhere. More generally one studies disintegration (or decomposition into slices) of a measure on a space relative to any mapping (instead of the projection , cf. [a11], [a12]).

For each bounded continuous function , let be the expectation of the random variable and let be the measure on . Then, using (a2), the disintegration of the Campbell measure on yields the measure on and, if is -finite, the can be normalized -almost everywhere to probability measures on to give

The are the Palm distributions (Palm probabilities) of . Equivalently, as a function of , for is -almost everywhere the Radon–Nikodým derivative (cf. Radon–Nikodým theorem) of the measure on with respect to . Here is the random measure ,

i.e. the trace of on .

#### References

[a1] | A.A. Borovkov, "Stochastic processes in queueing theory" , Springer (1976) (Translated from Russian) |

[a2] | P.A.W. Lewis (ed.) , Stochastic point processes: statistical analysis theory and applications , Wiley (Interscience) (1972) |

[a3] | V.K. Murthy, "The general point process" , Addison-Wesley (1974) |

[a4] | D.C. Snyder, "Random point processes" , Wiley (1975) |

[a5] | D.J. Daley, D. Vere-Jones, "An introduction to the theory of point processes" , Springer (1978) |

[a6] | F. Baccelli, P. Brémaud, "Palm probabilities and stationary queues" , Lect. notes in statistics , 41 , Springer (1987) |

[a7] | P. Brémaud, "Point processes and queues - Martingale dynamics" , Springer (1981) |

[a8] | J. Neveu, "Processus ponctuels" J. Hoffmann-Jørgensen (ed.) T.M. Liggett (ed.) J. Neveu (ed.) , Ecole d'été de St. Flour VI 1976 , Lect. notes in math. , 598 , Springer (1977) pp. 250–448 |

[a9] | O. Kallenberg, "Random measures" , Akademie Verlag & Acad. Press (1986) |

[a10] | J. Grandell, "Doubly stochastic Poisson processes" , Springer (1976) |

[a11] | H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) (Translated from German) |

[a12] | N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1967) pp. Chapt. 5: Intégration des mesures, §6.6 |

[a13] | N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1959) pp. Chapt. 6: Intégration vectorielle, §3 |

**How to Cite This Entry:**

Stochastic point process. Yu.K. Belyaev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Stochastic_point_process&oldid=16464