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Encyclopedia > Stochastic process

In the mathematics of probability, a stochastic process is a random function. In the most common applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field). Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This article does not cite its references or sources. ... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... Process (lat. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... Partial plot of a function f. ... In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space &#937;={0,1,...,G-1}n, a probability measure &#960; is a random field if . There exist several types of random fields, such as Markov...

Familiar examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material. The New York Stock Exchange A stock market is a market for the trading of company stock, and derivatives of same; both of these are securities listed on a stock exchange as well as those only traded privately. ... Look up Speech in Wiktionary, the free dictionary. ... Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... Video is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images which represent scenes in motion. ... Medicine is the branch of health science and the sector of public life concerned with maintaining or restoring human health through the study, diagnosis, treatment and possible prevention of disease and injury. ... Lead II An electrocardiogram (ECG or EKG, abbreviated from the German Elektrokardiogramm) is a graphic produced by an electrocardiograph, which records the electrical voltage in the heart in the form of a continuous strip graph. ... Electroencephalography is the neurophysiologic measurement of the electrical activity of the brain by recording from electrodes placed on the scalp or, in special cases, subdurally or in the cerebral cortex. ... A sphygmomanometer, a device used for measuring blood pressure. ... [[Image:Translational motion. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In mathematics and physics, a random walk, sometimes called a drunkards walk, is a formalisation of the intuitive idea of taking successive steps, each in a random direction. ... This article or section is in need of attention from an expert on the subject. ...

A stochastic process is a sequence of measurable functions, that is, a random variable X defined on a probability space (Ω, S, Pr) with values in a space of functions F. The space F in turn consists of functions ID. Thus a stochastic process can also be regarded as an indexed collection of random variables {Xi}, where the index i ranges through an index set I, defined on the probability space (Ω, S, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed collection of random variables is the most common one. A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, a probability space or probability measure is a set S, together with a Ïƒ-algebra X on S and a measure P on that Ïƒ-algebra such that P(S) = 1. ... In mathematics, an index set is another name for a function domain. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

A notable special case is where the index set is a discrete set I, often the nonnegative integers {0, 1, 2, 3, ...}. In mathematics, an index set is another name for a function domain. ...

In a continuous stochastic process the index set is continuous (usually space or time), resulting in an uncountably infinite number of random variables. In mathematics, a countable set is a set with the same cardinality (i. ...

Each point in the sample space Ω corresponds to a particular value for each of the random variables and the resulting function (mapping a point in the index set to the value of the random variable attached to it) is known as a realisation of the stochastic process. In the case the index family is a real (finite or infinite) interval, the resulting function is called a sample path. In probability theory, the sample space, often denoted S, Î© or U (for universe), of an experiment or random trial is the set of all possible outcomes. ...

A particular stochastic process is determined by specifying the joint probability distributions of the various random variables. Given two random variables X and Y, the joint distribution of X and Y is the distribution of X and Y together. ...

Stochastic processes may be defined in higher dimensions by attaching a multivariate random variable to each point in the index set, which is equivalent to using a multidimensional index set. Indeed a multivariate random variable can itself be viewed as a stochastic process with index set {1, ..., n}. A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Î©, P). ...

### Examples

The paradigm continuous stochastic process is that of the Wiener process. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and, since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus the random force is described by a two component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being homogeneous the force is independent of the spatial coordinates) with the domain of the two random variables being R, giving the x and y components of the force. A treatment of Brownian motion generally also includes the effect of viscosity, resulting in an equation of motion known as the Langevin equation. In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In statistical physics, a Langevin equation is a stochastic differential equation describing Brownian motion in a potential. ...

If the index set of the process is N (the natural numbers), and the range is R (the real numbers), there are some natural questions to ask about the sample sequences of a process {Xi}iN, where a sample sequence is {X(ω)i}iN. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

1. What is the probability that each sample sequence is bounded?
2. What is the probability that each sample sequence is monotonic?
3. What is the probability that each sample sequence has a limit as the index approaches ∞?
4. What is the probability that the series obtained from a sample sequence from f(i) converges?
5. What is the probability distribution of the sum?

Similarly, if the index space I is a finite or infinite interval, we can ask about the sample paths {X(ω)t}tI This article does not cite its references or sources. ... In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ...

1. What is the probability that it is bounded/integrable/continuous/differentiable...?
2. What is the probability that it has a limit at ∞
3. What is the probability distribution of the integral?

More Examples Integrability is a mathematical concept used in different areas. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

### Interesting special cases

• Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes, also called time-homogeneous.
• Bernoulli processes: discrete-time processes with two possible states.
• Bernoulli schemes: discrete-time processes with N possible states; every stationary process in N outcomes is a Bernoulli scheme, and vice-versa.
• Processes with independent increments: processes where the domain is at least partially ordered and, if $x_1 < ldots < x_n$, all the variables f(xk + 1) − f(xk) are independent. Markov chains are a special case.
• Markov processes are those in which the future is conditionally independent of the past given the present.
• Point processes: random arrangements of points in a space S. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of S, ordered by inclusion; the range is the set of natural numbers; and, if A is a subset of B, $f(A) le f(B)$ with probability 1.
• Gaussian processes: processes where all linear combinations of coordinates are normally distributed random variables.
• Poisson processes
• Gauss-Markov processes: processes that are both Gaussian and Markov
• Martingales -- processes with constraints on the expectation
• Galton-Watson processes
• Branching processes
• Gamma processes

### Finite-dimensional distributions and law

A great deal of information about a stochastic process can often be obtained from its finite-dimensional distributions (the measures induced on the finite Cartesian product of the state space at a finite sequence of times) and law (the measure induced on the collection of all functions from the index set into the state space). For sample continuous processes, the finite-dimensional distributions determine the law, and vice versa. A suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section). In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X Ã— Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after RenÃ© Descartes... In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. ... In mathematics, a sample continuous process is a stochastic process whose sample paths are almost surely continuous functions. ...

## Constructing stochastic processes

In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension. In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ... Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a &#963;-algebra (or &#963;-field) X over a set S is a family of subsets of S which is closed under countable set operations; &#963;-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a measure is a function that assigns a number, e. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (&#1040;&#1085;&#1076;&#1088;&#1077;&#769;&#1081; &#1053;&#1080;&#1082;&#1086;&#1083;&#1072;&#769;&#1077;&#1074;&#1080;&#1095; &#1050;&#1086;&#1083;&#1084;&#1086;&#1075;&#1086;&#769;&#1088;&#1086;&#1074;) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a...

There is at least one alternative axiomatization of probability theory by means of expectations on C-star algebras of random variables. In this case the method goes by the name of Gelfand-Naimark-Segal construction. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... C*-algebras are an important area of research in functional analysis. ... In the algebraic axiomatization of probability theory, one of whose main proponents was Irving Segal, the primary concept is not that of probability of an event, but rather that of a random variable. ... In functional analysis, given a C*-algebra A, the GNS construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A called states. ...

This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.

### The Kolmogorov extension

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions $f: X to Y$ exists, then it can be used to specify the probability distribution of finite-dimensional random variables $f(x_1),dots,f(x_n)$. Now, from this n-dimensional probability distribution we can deduce an (n − 1)-dimensional marginal probability distribution for $f(x_1),dots,f(x_{n-1})$. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation. In mathematics, a probability space is a set S, together with a &#963;-algebra X on S and a measure P on that &#963;-algebra such that P(S) = 1. ... In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. ...

The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition. In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably consistent collection of finite-dimensional distributions will define a stochastic process. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

### Separability, or what the Kolmogorov extension does not provide

Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer. In mathematics, a measure is a function that assigns a number, e. ...

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates [f(x1),...,f(xn)] are restricted to lie in measurable subsets of Yn. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer. In mathematics the term countable set is used to describe the size of a set, e. ...

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of calculus are of this sort. For example: Calculus is a central branch of mathematics. ...

all require knowledge of uncountably many values of the function. In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates {f(xi)} whose values determine the whole random function f. Separable can refer to: Separable space in topology Separable sigma algebra in measure theory Separable differential equations This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ...

The Kolmogorov continuity theorem guarantees that processes that satisfy certain constraints on the moments of their increments are continuous. In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constrains on the moments of its increments will be continuous (or, more precisely, have a continuous version). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov. ... -1...

In the mathematics of probability, a stochastic process can be thought of as a random function. ...

## References

Papoulis, Athanasios & Pillai, S. Unnikrishna (2001). Probability, Random Variables and Stochastic Processes. McGraw-Hill Science/Engineering/Math. ISBN 0-07-281725-9.

Results from FactBites:

 Stochastic process - Wikipedia, the free encyclopedia (1378 words) A stochastic process is a random function, that is a random variable X defined on a probability space (Ω, Pr) with values in a space of functions F. Stochastic processes may be defined in higher dimensions by attaching a multivariate random variable to each point in the index set, which is equivalent to using a multidimensional index set. The paradigm continuous stochastic process is that of the Wiener process.
 NationMaster - Encyclopedia: Stochastic process (3499 words) A point process is a type of stochastic process that is widely used in many fields of applied mathematics, such as queueing theory and computational neuroscience. The Galton-Watson process is a stochastic process arising from Francis Galtons statistical investigation of the extinction of surnames. In statistics, a stochastic process is often known as a time series, where the index set is a finite (or at most countable) ordered sequence of real numbers.
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