P. BREMAUD, CEREMADE, Universite de Paris IX (Dauphine). Abstract Optimal stochastic control of point processes (and more generally of marked. Increas- ingly, spatial-temporal point processes are used to describe environmental process. This sort of definition is used by Jacod (), Brémaud (). Authors; Authors and affiliations. P. Bremaud Point Process Counting Process Jump Process Stochastic Integration Local Martingale. These keywords were.
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In probability theory and statisticsCampbell’s theorem or the Campbell—Hardy theorem is either a particular equation or procdsses of results relating to the expectation of a function summed over a point process to an integral ;oint the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum.
One version of the theorem,  also known as Campbell’s formula: All these results are employed in probability and statistics with a particular importance in the theory of point processes  and queueing theory  as well as the related fields stochastic geometry continuum percolation theory and spatial statistics.
Another result by the name of Campbell’s theorem  is specifically for proceswes Poisson point process and gives a method for calculating moments as well as the Laplace functional of a Poisson point process.
The name of both theorems stems from the work   by Norman R.
A Martingale Approach to Point Processes – Pierre Brémaud – Google Books
Campbell on thermionic noise, also known as shot noisein vacuum tubes  which was partly inspired by the work of Ernest Rutherford and Hans Geiger on alpha particle detection, where the Poisson point process arose as a solution to a family of differential equations by Ponit Bateman. Hardywhich has inspired the result to be sometimes called the Campbell—Hardy theorem. This equation naturally holds for the homogeneous Poisson point processes, which is an example of a stationary stochastic process.
Campbell’s theorem for general point processes gives a method for calculating the expectation of a function of a point of a point process summed over all the points in the point process. These random sums over point processes have applications in many areas where they are used as mathematical models.
Campbell originally studied a problem of random sums motivated by understanding thermionic noise in valves, which is also known as shot-noise. Consequently, the study of brrmaud sums of functions over point processes is known as shot noise in probability and, particularly, point process theory.
In wireless network communication, when a transmitter is trying to send a signal to a receiver, all the other transmitters in the network can be considered as interference, which poses a similar problem as noise does in traditional wired telecommunication networks in terms of the ability to send data based on information theory. If the positioning of the interfering ppint are assumed to form some point process, then shot noise can be used to model the sum of their interfering signals, which has led to stochastic geometry models of wireless networks.
For general point processes, other more general versions gremaud Campbell’s theorem exist depending on the nature of the random sum and in particular the function being summed over the point process. If the function is a function of more than one point of the point process, the moment measures or factorial moment measures of the point process are needed, which can be compared to moments and factorial of random variables.
The type of pocesses needed depends on whether the points of the point process in the random sum are need to be distinct or may repeat.
Factorial moment measures are used when points are not allowed to repeat, hence points are distinct. For general point processes, Campbell’s theorem is only for sums of functions of a single point of the point process.
To calculate poitn sum of a function of a single point as well as the entire point process, then generalized Campbell’s theorems are required using the Palm distribution of the point process, which is based on the branch of probability known as Palm theory or Palm calculus. From this theorem some expectation results for the Poisson point process follow, including its Laplace functional.
From Wikipedia, the free encyclopedia. This article is about random point processes. For other uses, see Campbell’s theorem geometry. Stochastic procseses and its applicationsvolume 2.
Lecture Notes in Mathematics. Probability and its Applications.
Elements of queueing theory: Palm Martingale calculus and stochastic recurrences. Continuum percolation, volume of Cambridge tracts in mathematics, Probability and random processes. Probability and Its Applications. Fourier Analysis of Stochastic Processes. A crash course in stochastic geometry.