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
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 ...
, Campbell's theorem offers a way to calculate expectations of a real-valued function
defined also on
and summed over
, namely:
:
where
denotes the expectation and set notation is used such that
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
, Campbell's theorem relates the above expectation with the intensity measure
. 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
is defined as:
:
where the
measure notation is used such that
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
can be interpreted as the average number of points of the point process
located in the set ''B''.
First definition: general point process
One version of Campbell's theorem for a general (not necessarily simple) point process
with intensity measure:
:
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
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
and a measurable function
, the sum of
over the point process is given by the equation:
:
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:
:
If the intensity measure
of a point process
has a density
, then Campbell's formula becomes:
:
Stationary point process
For a stationary point process
with constant density
, Campbell's theorem or formula reduces to a volume integral:
:
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
:
In this case the cumulants
of
equal
:
where
are the raw moments of the distribution of
.
[
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
with intensity measure
and a measurable function
, the random sum
:
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
:
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
the equation
:
holds if the integral on the right-hand side
converges, which is the case for purely
imaginary . Moreover,
:
and if this integral converges, then
:
where
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
.
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
with intensity measure
, 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:
:
which for the homogeneous case is:
:
Notes
References
{{reflist, 29em
Theorems in probability theory