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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### Campbell’s theorem (probability) – Wikipedia

This equation naturally holds for the homogeneous Poisson point processes, which is an example of a stationary stochastic process. Facebook Twitter Advertising and Corporate Services. Views Read Edit View history. For general point processes, other more general versions of Campbell’s theorem exist depending on the nature of the random sum and in particular the function being summed over the point process.

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Close mobile search navigation Article navigation. Read, highlight, and take notes, across web, tablet, and phone. The type of measure needed depends on whether the points of the point process in the random sum are need to be distinct or may repeat.

Another result by the name of Campbell’s theorem [7] is specifically for the Poisson point process and gives a method for calculating moments as well as the Laplace functional of a Poisson point process. Probability and random processes. Chapman and Hall pppages 1—35, From Wikipedia, the free encyclopedia. A crash course in stochastic geometry.

Issues About Advertising and Corporate Services. My library Help Advanced Book Search. Retrieved from ” https: 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. Campbell on thermionic noise, also known as shot noisein vacuum tubes[3] [10] which was partly inspired by the work of Ernest Rutherford and Hans Procewses on alpha particle detection, where the Poisson point process arose as a solution to loint family of differential equations by Harry Bateman.

## Pierre Brémaud

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.

Oxford Processses Press is a department of the University of Oxford. Likelihood Ratios and Martingale. For general point processes, Campbell’s theorem is only for sums of functions of a single point of the point process.

### CiteSeerX — Hawkes Branching Point Processes without Ancestors

Citing articles via Google Scholar. Palm Martingale calculus and stochastic recurrences. Fourier Analysis of Stochastic Processes.

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This page was last edited on 22 Julyat Stochastic geometry and its applicationsvolume 2.

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Related articles in Google Scholar. The name of both theorems stems from the work [8] [9] by Norman R. Account Options Sign in. From this theorem some expectation results for the Poisson point process follow, including its Laplace functional. In probability theory and statisticsCampbell’s theorem or the Campbell—Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum.

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Probability and Its Applications. Don’t bremaaud an account? Campbell originally studied a problem of random sums motivated by understanding thermionic noise in valves, which is also known as shot-noise.

This article is about random point processes. Article PDF first page preview. If the bemaud of the interfering transmitters 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.