![BMonSphere](https://upload.wikimedia.org/wikipedia/commons/4/4e/BMonSphere.jpg)
In
probability theory
Probability theory 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 expressing it through a set o ...
and related fields, a stochastic () or random process is a
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical pr ...
usually defined as a
family
Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. Stochastic processes are widely used as
mathematical models
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
of systems and phenomena that appear to vary in a random manner. Examples include the growth of a
bacteria
Bacteria (; singular: bacterium) are ubiquitous, mostly free-living organisms often consisting of one biological cell. They constitute a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria were among ...
l population, an
electrical current
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
fluctuating due to
thermal noise
A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
, or the movement of a
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
.
Stochastic processes have applications in many disciplines such as
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
,
ecology
Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...
,
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
,
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
,
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
and
telecommunications
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
.
Furthermore, seemingly random changes in
financial market
A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets ...
s have motivated the extensive use of stochastic processes in
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
.
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
or Brownian motion process, used by
Louis Bachelier
Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part ...
to study price changes on the
Paris Bourse
Euronext Paris is France's securities market, formerly known as the Paris Bourse, which merged with the Amsterdam, Lisbon, and Brussels exchanges in September 2000 to form Euronext NV. As of 2022, the 795 companies listed had a combined market ...
,
and the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
, used by
A. K. Erlang
Agner Krarup Erlang (1 January 1878 – 3 February 1929) was a Denmark, Danish mathematician, statistician and engineer, who invented the fields of teletraffic engineering, traffic engineering and queueing theory.
By the time of his relatively ...
to study the number of phone calls occurring in a certain period of time.
These two stochastic processes are considered the most important and central in the theory of stochastic processes,
and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.
The term random function is also used to refer to a stochastic or random process,
because a stochastic process can also be interpreted as a random element in a
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
.
The terms ''stochastic process'' and ''random process'' are used interchangeably, often with no specific
mathematical space
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, i ...
for the set that indexes the random variables.
But often these two terms are used when the random variables are indexed by the
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
or an
interval of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
.
If the random variables are indexed by the
Cartesian plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
or some higher-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, then the collection of random variables is usually called a
random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other d ...
instead.
The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
s,
martingales,
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
es,
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
es,
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
es, random fields,
renewal process
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ho ...
es, and
branching process
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. The random variables of a stochastic process are indexed by the natural numbers. The origi ...
es.
The study of stochastic processes uses mathematical knowledge and techniques from
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
,
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
,
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
,
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, and
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
as well as branches of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
such as
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
,
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
,
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
, and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
The theory of stochastic processes is considered to be an important contribution to mathematics
and it continues to be an active topic of research for both theoretical reasons and applications.
Introduction
A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.
The set used to index the random variables is called the index set. Historically, the index set was some
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, such as the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
, giving the index set the interpretation of time.
Each random variable in the collection takes values from the same
mathematical space
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, i ...
known as the state space. This state space can be, for example, the integers, the real line or
-dimensional Euclidean space.
An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time.
A stochastic process can have many
outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization.
Classifications
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the index set and the state space.
When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in
discrete time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
.
If the index set is some interval of the real line, then time is said to be
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. The two types of stochastic processes are respectively referred to as discrete-time and
continuous-time stochastic process In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time pr ...
es.
Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.
If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence.
If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is
-dimensional Euclidean space, then the stochastic process is called a
-dimensional vector process or
-vector process.
Etymology
The word ''stochastic'' in
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
word meaning "to aim at a mark, guess", and the
Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a com ...
gives the year 1662 as its earliest occurrence.
In his work on probability ''Ars Conjectandi'', originally published in Latin in 1713,
Jakob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".
This phrase was used, with reference to Bernoulli, by
Ladislaus Bortkiewicz
Ladislaus Josephovich Bortkiewicz (Russian Владислав Иосифович Борткевич, German ''Ladislaus von Bortkiewicz'' or ''Ladislaus von Bortkewitsch'') (7 August 1868 – 15 July 1931) was a Russian economist and statisti ...
who in 1917 wrote in German the word ''stochastik'' with a sense meaning random. The term ''stochastic process'' first appeared in English in a 1934 paper by
Joseph Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, ...
.
For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term ''stochastischer Prozeß'' was used in German by
Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to th ...
,
though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.
According to the Oxford English Dictionary, early occurrences of the word ''random'' in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term ''random process'' pre-dates ''stochastic process'', which the Oxford English Dictionary also gives as a synonym, and was used in an article by
Francis Edgeworth
Francis Ysidro Edgeworth (8 February 1845 – 13 February 1926) was an Anglo-Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward, he was appointed the ...
published in 1888.
Terminology
The definition of a stochastic process varies,
but a stochastic process is traditionally defined as a collection of random variables indexed by some set.
The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.
Both "collection",
or "family" are used
while instead of "index set", sometimes the terms "parameter set"
or "parameter space"
are used.
The term ''random function'' is also used to refer to a stochastic or random process,
though sometimes it is only used when the stochastic process takes real values.
This term is also used when the index sets are mathematical spaces other than the real line,
[, p. 1] while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,
and other terms are used such as ''random field'' when the index set is
-dimensional Euclidean space
or a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
.
Notation
A stochastic process can be denoted, among other ways, by
,
,
[,] or simply as
or
, although
is regarded as an
abuse of function notation.
For example,
or
are used to refer to the random variable with the index
, and not the entire stochastic process.
If the index set is
, then one can write, for example,
to denote the stochastic process.
Examples
Bernoulli process
One of the simplest stochastic processes is the Bernoulli process,
which is a sequence of
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
(iid) random variables, where each random variable takes either the value one or zero, say one with probability
and zero with probability
. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is
and its value is one, while the value of a tail is zero.
In other words, a Bernoulli process is a sequence of iid Bernoulli random variables,
where each coin flip is an example of a
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
.
Random walk
Random walks
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
are stochastic processes that are usually defined as sums of
iid
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independence (probability theory), ...
random variables or random vectors in Euclidean space, so they are processes that change in discrete time.
But some also use the term to refer to processes that change in continuous time,
particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism.
There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.
A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say,
, or decreases by one with probability
, so the index set of this random walk is the natural numbers, while its state space is the integers. If the
, this random walk is called a symmetric random walk.
Wiener process
The Wiener process is a stochastic process with
stationary and
independent increments In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochas ...
that are
normally distributed based on the size of the increments.
The Wiener process is named after
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for
Brownian movement in liquids.
![DriftedWienerProcess1D](https://upload.wikimedia.org/wikipedia/commons/3/3f/DriftedWienerProcess1D.svg)
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.
Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.
But the process can be defined more generally so its state space can be
-dimensional Euclidean space.
If the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant
, which is a real number, then the resulting stochastic process is said to have drift
.
Almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
, a sample path of a Wiener process is continuous everywhere but
nowhere differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. It can be considered as a continuous version of the simple random walk.
The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,
which is the subject of
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem.
Let X_1, X_2, X_3, \ldots be ...
or invariance principle, also known as the functional central limit theorem.
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.
The process also has many applications and is the main stochastic process used in stochastic calculus.
It plays a central role in quantitative finance,
where it is used, for example, in the Black–Scholes–Merton model.
The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.
Poisson process
The Poisson process is a stochastic process that has different forms and definitions.
It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.
The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.
If the parameter constant of the Poisson process is replaced with some non-negative integrable function of
, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.
Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.
Defined on the real line, the Poisson process can be interpreted as a stochastic process,
among other random objects.
But then it can be defined on the
-dimensional Euclidean space or other mathematical spaces,
where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.
In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.
But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
Definitions
Stochastic process
A stochastic process is defined as a collection of random variables defined on a common
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, where
is a
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
,
is a
-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, and
is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
; and the random variables, indexed by some set
, all take values in the same mathematical space
, which must be
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
with respect to some
-algebra
.
In other words, for a given probability space
and a measurable space
, a stochastic process is a collection of
-valued random variables, which can be written as:
Historically, in many problems from the natural sciences a point
had the meaning of time, so
is a random variable representing a value observed at time
.
A stochastic process can also be written as
to reflect that it is actually a function of two variables,
and
.
There are other ways to consider a stochastic process, with the above definition being considered the traditional one.
For example, a stochastic process can be interpreted or defined as a
-valued random variable, where
is the space of all the possible
functions from the set
into the space
.
However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined.
Index set
The set
is called the index set
or parameter set
of the stochastic process. Often this set is some subset of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, such as the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
or an interval, giving the set
the interpretation of time.
In addition to these sets, the index set
can be another set with a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
or a more general set,
such as the Cartesian plane
or
-dimensional Euclidean space, where an element
can represent a point in space.
That said, many results and theorems are only possible for stochastic processes with a totally ordered index set.
State space
The
mathematical space
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, i ...
of a stochastic process is called its ''state space''. This mathematical space can be defined using
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s,
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
s,
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
Sample function
A sample function is a single
outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.
More precisely, if
is a stochastic process, then for any point
, the
mapping
is called a sample function, a realization, or, particularly when
is interpreted as time, a sample path of the stochastic process
.
This means that for a fixed
, there exists a sample function that maps the index set
to the state space
.
Other names for a sample function of a stochastic process include trajectory, path function
or path.
Increment
An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if
is a stochastic process with state space
and index set
, then for any two non-negative numbers
and
such that
, the difference
is a
-valued random variable known as an increment.
When interested in the increments, often the state space
is the real line or the natural numbers, but it can be
-dimensional Euclidean space or more abstract spaces such as
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s.
Further definitions
Law
For a stochastic process
defined on the probability space
, the law of stochastic process
is defined as the
image measure:
where
is a probability measure, the symbol
denotes function composition and
is the pre-image of the measurable function or, equivalently, the
-valued random variable
, where
is the space of all the possible
-valued functions of
, so the law of a stochastic process is a probability measure.
For a measurable subset
of
, the pre-image of
gives
so the law of a
can be written as:
The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution.
Finite-dimensional probability distributions
For a stochastic process
with law
, its finite-dimensional distribution for
is defined as:
This measure
is the joint distribution of the random vector
; it can be viewed as a "projection" of the law
onto a finite subset of
.
For any measurable subset
of the
-fold
Cartesian power
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
, the finite-dimensional distributions of a stochastic process
can be written as:
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.
Stationarity
Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if
is a stationary stochastic process, then for any
the random variable
has the same distribution, which means that for any set of
index set values
, the corresponding
random variables
all have the same
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.
But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.
When the index set
can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.
The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.
A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.
A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process
is said to be stationary in the wide sense, then the process
has a finite second moment for all
and the covariance of the two random variables
and
depends only on the number
for all
.
Khinchin
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to th ...
introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.
Filtration
A
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration
, on a probability space
is a family of sigma-algebras such that
for all
, where
and
denotes the total order of the index set
.
With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process
at
, which can be interpreted as time
.
The intuition behind a filtration
is that as time
passes, more and more information on
is known or available, which is captured in
, resulting in finer and finer partitions of
.
Modification
A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process
that has the same index set
, state space
, and probability space
as another stochastic process
is said to be a modification of
if for all
the following
holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law
and they are said to be stochastically equivalent or equivalent.
Instead of modification, the term version is also used,
however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the
Kolmogorov continuity theorem In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is ...
says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.
The theorem can also be generalized to random fields so the index set is
-dimensional Euclidean space
as well as to stochastic processes with
metric spaces
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
as their state spaces.
Indistinguishable
Two stochastic processes
and
defined on the same probability space
with the same index set
and set space
are said be indistinguishable if the following
holds.
If two
and
are modifications of each other and are almost surely continuous, then
and
are indistinguishable.
Separability
Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a
separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the ...
, which means that the index set has a dense countable subset.
More precisely, a real-valued continuous-time stochastic process
with a probability space
is separable if its index set
has a dense countable subset
and there is a set
of probability zero, so
, such that for every open set
and every closed set
, the two events
and
differ from each other at most on a subset of
.
The definition of separability can also be stated for other index sets and state spaces,
[, p. 22] such as in the case of random fields, where the index set as well as the state space can be
-dimensional Euclidean space.
The concept of separability of a stochastic process was introduced by
Joseph Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, ...
,.
The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.
Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.
A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.
Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.
Independence
Two stochastic processes
and
defined on the same probability space
with the same index set
are said be independent if for all
and for every choice of epochs
, the random vectors
and
are independent.
[Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.]
Uncorrelatedness
Two stochastic processes
and
are called uncorrelated if their cross-covariance