HOME

TheInfoList



OR:

In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, Campbell's theorem or the Campbell–Hardy theorem is either a particular
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
or set of results relating to the expectation of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
summed over a
point process In statistics and probability theory, a point process or point field is a set of a random number of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', ...
to an
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
involving the mean measure of the point process, which allows for the calculation of
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the
random In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
sum. One version of the theorem,D. Stoyan, W. S. Kendall, J. Mecke. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995. also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving
moment measure In probability and statistics, a moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as ...
s and
factorial moment measure In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes ofte ...
s that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of
point process In statistics and probability theory, a point process or point field is a set of a random number of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', ...
es and
queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because th ...
as well as the related fields
stochastic geometry In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
,
continuum percolation theory In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ). More specifically, the underlying points of discrete percolation fo ...
,R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge tracts in mathematics, 1996. and
spatial statistics Spatial statistics is a field of applied statistics dealing with spatial data. It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional ( areal unit) and lattice ( gridded) data, poin ...
. Another result by the name of Campbell's theorem is specifically for the
Poisson point process 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 ...
and gives a method for calculating moments as well as the
Laplace functional In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properti ...
of a Poisson point process. The name of both theorems stems from the work by Norman R. Campbell on
thermionic Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
noise, also known as
shot noise Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
, in
vacuum tubes A vacuum tube, electron tube, thermionic valve (British usage), or tube (North America) is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied. It ...
, which was partly inspired by the work of
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson (30 August 1871 – 19 October 1937) was a New Zealand physicist who was a pioneering researcher in both Atomic physics, atomic and nuclear physics. He has been described as "the father of nu ...
and
Hans Geiger Johannes Wilhelm Geiger ( , ; ; 30 September 1882 – 24 September 1945) was a German nuclear physicist. He is known as the inventor of the Geiger counter, a device used to detect ionizing radiation, and for carrying out the Rutherford scatt ...
on
alpha particle Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay but may also be produce ...
detection, where the
Poisson point process 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 ...
arose as a solution to a family of differential equations by
Harry Bateman Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poinca ...
. In Campbell's work, he presents the moments and
generating functions In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, which has inspired the result to be sometimes called the Campbell–Hardy theorem.


Background

For a point process N defined on ''d''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\textbf^d , Campbell's theorem offers a way to calculate expectations of a real-valued function f defined also on \textbf^d and summed over N, namely: : \operatorname E\left \sum_f(x)\right where E denotes the expectation and set notation is used such that N is considered as a random set (see
Point process notation In probability and statistics, point process notation comprises the range of mathematical notation used to symbolically represent random objects known as point processes, which are used in related fields such as stochastic geometry, spatial sta ...
). For a point process N, Campbell's theorem relates the above expectation with the intensity measure \Lambda. In relation to a
Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
''B'' the intensity measure of N is defined as: :\Lambda(B)=\operatorname E (B) where the measure notation is used such that N is considered a random
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
. The quantity \Lambda(B) can be interpreted as the average number of points of the point process N located in the set ''B''.


First definition: general point process

One version of Campbell's theorem for a general (not necessarily simple) point process N with intensity measure: : \Lambda (B)=\operatorname E (B) is known as Campbell's formula or Campbell's theorem,P. Brémaud. ''Fourier Analysis of Stochastic Processes''. Springer, 2014. which gives a method for calculating expectations of sums of
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s f with
ranges In the Hebrew Bible and in the Old Testament, the word ranges has two very different meanings. Leviticus In Leviticus 11:35, ranges (כירים) probably means a cooking furnace for two or more pots, as the Hebrew word here is in the dual numbe ...
on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. More specifically, for a point process N and a measurable function f: \textbf^d\rightarrow \textbf, the sum of f over the point process is given by the equation: : E\left sum_f(x)\right\int_ f(x)\Lambda (dx), where if one side of the equation is finite, then so is the other side.A. Baddeley. A crash course in stochastic geometry. ''Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall) pp'', pages 1–35, 1999. This equation is essentially an application of
Fubini's theorem In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is L ...
and it holds for a wide class of point processes, simple or not. Depending on the integral notation, this integral may also be written as: : \operatorname E\left sum_f(x)\right\int_ f \, d\Lambda , If the intensity measure \Lambda of a point process N has a density \lambda(x) , then Campbell's formula becomes: : \operatorname E\left sum_f(x)\right \int_ f(x)\lambda(x) \, dx


Stationary point process

For a stationary point process N with constant density \lambda>0, Campbell's theorem or formula reduces to a volume integral: : \operatorname E\left sum_f(x)\right\lambda \int_ f(x) \, dx This equation naturally holds for the homogeneous Poisson point processes, which is an example of a
stationary stochastic process In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...
.


Applications: Random sums

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.


Shot noise

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 random sums of functions over point processes is known as shot noise in probability and, particularly, point process theory.


Interference in wireless networks

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 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.


Neuroscience

The total input in neurons is the sum of many synaptic inputs with similar time courses. When the inputs are modeled as independent Poisson point process, the mean current and its variance are given by Campbell theorem. A common extension is to consider a sum with random amplitudes : S =\sum_ a_n f(x) In this case the cumulants \kappa_i of S equal : \kappa_i= \lambda \overline \int f(x)^i dx where \overline are the raw moments of the distribution of a . S.O. Rice ''Mathematical analysis of random noise'' Bell Syst. Tech. J. 24, 1944 reprinted in "'Selected papers on noise and random processes'' N. Wax (editor) Dover 1954.


Generalizations

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.


Functions of multiple points

If the function is a function of more than one point of the point process, the
moment measure In probability and statistics, a moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as ...
s or
factorial moment measure In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes ofte ...
s of the point process are needed, which can be compared to moments and factorial of random variables. 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.


Repeating points

Moment measures are used when points are allowed to repeat.


Distinct points

Factorial moment measures are used when points are not allowed to repeat, hence points are distinct.


Functions of points and the point process

For general point processes, Campbell's theorem is only for sums of functions of a single point of the point process. To calculate the 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.


Second definition: Poisson point process

Another version of Campbell's theorem says that for a Poisson point process N with intensity measure \Lambda and a measurable function f:\textbf^d\rightarrow \textbf, the random sum : S =\sum_f(x) is
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
with probability one
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the integral : \int_ \min(, f(x), ,1)\Lambda (dx) < \infty. Provided that this integral is finite, then the theorem further asserts that for any
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
value \theta the equation : E(e^)=\exp \left(\int_ ^-1Lambda (dx) \right), holds if the integral on the right-hand side converges, which is the case for purely imaginary \theta. Moreover, : E(S)=\int_ f(x)\Lambda (dx), and if this integral converges, then : \operatorname(S)=\int_ f(x)^2\Lambda (dx), where \text(S) denotes the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the random sum S. From this theorem some expectation results for the
Poisson point process 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 ...
follow, including its
Laplace functional In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properti ...
.


Application: Laplace functional

For a Poisson point process N with intensity measure \Lambda, the
Laplace functional In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properti ...
is a consequence of the above version of Campbell's theorem and is given by: : \mathcal_N(sf) := E\bigl e^ \bigr=\exp \Bigl \int_ (1-e^)\Lambda(dx) \Bigr which for the homogeneous case is: : \mathcal_N(sf)=\exp\Bigl \lambda\int_(1-e^) \, dx \Bigr


Notes


References

{{reflist, 29em Theorems in probability theory