Wikia

Psychology Wiki

Changes: Stochastic process

Edit

Back to page

 
(update wp)
 
Line 1: Line 1:
 
{{StatsPsy}}
 
{{StatsPsy}}
In the [[mathematics]] of [[probability]], a '''[[stochastic]] [[process]]''' is a [[random variable|random]] [[function (mathematics)|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]]).
+
In [[probability theory]], a '''stochastic process''', or sometimes '''random process''', is the counterpart to a deterministic process (or [[deterministic system]]). Instead of dealing with only one possible reality of how the process might evolve under time (as is the case, for example, for solutions of an [[ordinary differential equation]]), in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths may be more probable and others less.
   
Familiar examples of processes modeled as stochastic time series include [[stock market]] and [[exchange rate]] fluctuations, signals such as [[speech]], [[sound|audio]] and [[video]]; [[medicine|medical]] data such as a patient's [[Electrocardiogram|EKG]], [[Electroencephalography|EEG]], [[blood pressure]] or [[temperature]]; and random movement such as [[Brownian motion]] or [[random walk]]s. Examples of random fields include static images, random [[topography|topographies]] (landscapes), or composition variations of an inhomogeneous material.
+
In the simplest possible case ([[discrete-time|discrete time]]), a stochastic process amounts to a [[sequence (mathematics)|sequence]] of random variables known as a [[time series]] (for example, see [[Markov chain]]). Another basic type of a stochastic process is a [[random field]], whose domain is a region of [[space]], in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as [[function (mathematics)|function]]s of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are [[random variables]]: non-deterministic (single) quantities which have certain [[probability distribution]]s. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same type.<ref>Mathematically speaking, the type refers to the [[codomain]] of the function.</ref> Although the random values of a stochastic process at different times may be [[statistical independence|independent random variables]], in most commonly considered situations they exhibit complicated statistical correlations.
   
== Definition ==
+
Familiar examples of processes modeled as stochastic time series include signals such as [[Speech communication|speech]], [[sound|audio]] and [[video]], [[medicine|medical]] data such as a patient's [[Electrocardiogram|EKG]], [[Electroencephalography|EEG]], [[blood pressure]] or [[temperature]]. Examples of random fields include static images,
   
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 ''I'' → ''D''. Thus a stochastic process can also be regarded as
+
== Formal definition and basic properties ==
an indexed collection of random variables {'''X'''<sub>''i''</sub>}, 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 number]]s '''R'''). This view of a stochastic process as an indexed collection of random variables is the most common one.
+
=== Definition ===
  +
Given a [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>,
  +
a '''stochastic process''' (or '''random process''') with state space ''X'' is a collection of ''X''-valued
  +
[[random variable]]s indexed by a set ''T'' ("time"). That is, a stochastic process ''F'' is a collection
  +
: <math> \{ F_t : t \in T \}</math>
  +
where each <math>F_t</math> is an ''X''-valued random variable.
   
A notable special case is where the [[index set]] is a discrete set ''I'', often the nonnegative integers {0, 1, 2, 3, ...}.
+
A '''modification''' ''G'' of the process ''F'' is a stochastic process on the same state space, with the same parameter set ''T'' such that
  +
:<math> P ( F_t = G_t) =1 \qquad \forall t \in T.</math>
  +
A modification is '''indistinguishable''' if
  +
:<math> P ( \forall t \in T \, F_t = G_t) =1 .</math>
   
In a continuous stochastic process the index set is continuous (usually space or time), resulting in an [[countable set|uncountably infinite]] number of random variables.
+
=== Finite-dimensional distributions ===
   
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.
+
Let ''F'' be an ''X''-valued stochastic process. For every finite subset <math>T' \subseteq T</math>, we may write
  +
<math>T'=\{ t_1, \ldots, t_k \}</math>, where <math>k=\left|T'\right|</math> and the restriction <math>F|_{T'}=(F_{t_1}, F_{t_2},\ldots, F_{t_k})</math> is a random variable taking values in <math>X^k</math>. The distribution <math>\mathbb{P}_{T'}= \mathbb{P} (F|_{T'})^{-1}</math> of this random variable is a probability measure on <math>X^k</math>.
  +
Such random variables are called the [[finite-dimensional distribution]]s of ''F''.
   
A particular stochastic process is determined by specifying the [[joint probability distribution]]s of the various random variables.
+
Under suitable topological restrictions, a suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).
   
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''}.
+
== Construction ==
 
=== 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 [[wiktionary:Homogeneous|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]].
 
 
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 {'''X'''<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>, where a sample sequence is
 
{'''X'''(ω)<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>.
 
 
# What is the [[probability]] that each sample sequence is [[bounded function|bounded]]?
 
# What is the probability that each sample sequence is [[monotonic]]?
 
# What is the probability that each sample sequence has a [[Limit (mathematics)|limit]] as the index approaches ∞?
 
# What is the probability that the [[series (mathematics)|series]] obtained from a sample sequence from <math>f(i)</math> [[convergence|converges]]?
 
# What is the probability [[probability distribution|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)<sub>''t''</sub>}<sub>''t '' ∈ ''I''</sub>
 
# What is the probability that it is bounded/[[integrable]]/[[continuous function|continuous]]/[[differentiable]]...?
 
# What is the probability that it has a limit at ∞
 
# What is the probability distribution of the integral?
 
 
[http://www.hanoivn.net More Examples]
 
 
=== Interesting special cases ===
 
 
*[[Homogeneous process]]es: processes where the domain has some [[symmetry]] and the finite-dimensional probability distributions also have that symmetry. Special cases include [[stationary process]]es, also called time-homogeneous.
 
*[[Bernoulli process]]es: [[discrete-time]] processes with two possible states.
 
*[[Bernoulli scheme]]s: discrete-time processes with ''N'' possible states; every stationary process in ''N'' outcomes is a Bernoulli scheme, and vice-versa.
 
*[[process with independent increments|Processes with independent increments]]: processes where the domain is at least partially ordered and, if <math>x_1 < \ldots < x_n</math>, all the variables <math>f(x_k+1) - f(x_k)</math> are independent. [[Markov chain]]s are a special case.
 
*See also [[continuous-time Markov process]].
 
*[[Markov process]]es are those in which the future is conditionally independent of the past given the present.
 
*[[Point process]]es: random arrangements of points in a space <math>S</math>. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of <math>S</math>, ordered by inclusion; the range is the set of natural numbers; and, if A is a subset of B, <math>f(A) \le f(B)</math> with probability 1.
 
*[[Gaussian process]]es: processes where all linear combinations of coordinates are [[normal distribution|normally distributed]] random variables.
 
*[[Poisson process]]es
 
*[[Gauss-Markov process]]es: processes that are both Gaussian and Markov
 
*[[Martingale (probability theory)|Martingale]]s -- processes with constraints on the expectation
 
*[[Galton-Watson process]]es
 
*[[Elevator paradox]]
 
*[[Branching process]]es
 
*[[Gamma process]]es
 
 
*Many stochastic processes are [[Lévy process]]es.
 
 
===Finite-dimensional distributions and law===
 
 
A great deal of information about a stochastic process can often be obtained from its [[finite-dimensional distribution]]s (the measures induced on the finite [[Cartesian product]] of the state space at a finite sequence of times) and [[Law (stochastic processes)|law]] (the measure induced on the collection of all functions from the index set into the state space). For [[sample continuous process]]es, 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).
 
 
== 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 set|measurable subsets]] of the space of all functions, and then put a finite [[Measure (mathematics)|measure]] on it. For this purpose one traditionally uses a method called [[Kolmogorov]] extension.
 
In the ordinary [[axiomatization]] of [[probability theory]] by means of [[measure theory]], the problem is to construct a [[sigma-algebra]] of [[measurable set|measurable subsets]] of the space of all functions, and then put a finite [[Measure (mathematics)|measure]] on it. For this purpose one traditionally uses a method called [[Kolmogorov]] extension.
Line 25: Line 25:
 
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.
 
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 ===
+
=== Kolmogorov extension ===
   
The Kolmogorov extension proceeds along the following lines: assuming that a [[probability measure]] on the space of all functions <math>f: X \to Y</math> exists, then it can be used to specify the probability distribution of finite-dimensional random variables <math>f(x_1),\dots,f(x_n)</math>. Now, from this ''n''-dimensional probability distribution we can deduce an (''n''&nbsp;&minus;&nbsp;1)-dimensional [[marginal probability distribution]] for <math>f(x_1),\dots,f(x_{n-1})</math>. 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 density function|probability densities]], the result is called the [[Chapman-Kolmogorov equation]].
+
The Kolmogorov extension proceeds along the following lines: assuming that a [[probability measure]] on the space of all functions <math>f: X \to Y</math> exists, then it can be used to specify the joint probability distribution of finite-dimensional random variables <math>f(x_1),\dots,f(x_n)</math>. Now, from this ''n''-dimensional probability distribution we can deduce an (''n''&nbsp;&minus;&nbsp;1)-dimensional [[marginal probability distribution]] for <math>f(x_1),\dots,f(x_{n-1})</math>. Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the full-blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a [[Wiener process]] (in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. When this condition is expressed in terms of [[probability density function|probability densities]], the result is called the [[Chapman–Kolmogorov equation]].
   
 
The [[Kolmogorov extension theorem]] guarantees the existence of a stochastic process with a given family of finite-dimensional [[probability distribution]]s satisfying the Chapman-Kolmogorov compatibility condition.
 
The [[Kolmogorov extension theorem]] guarantees the existence of a stochastic process with a given family of finite-dimensional [[probability distribution]]s satisfying the Chapman-Kolmogorov compatibility condition.
Line 33: Line 33:
 
=== Separability, or what the Kolmogorov extension does not provide ===
 
=== 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.
+
Recall that in the Kolmogorov [[Axiomatic system|axiomatization]], [[measurable]] sets are the sets which have a probability or, in other words, the sets corresponding to [[yes-no question|yes/no question]]s that have a probabilistic answer.
   
The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates <math>[f(x_1), ..., f(x_n)]</math> are restricted to lie in measurable subsets of <math>Y_n</math>. 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.
+
The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates <math>[f(x_1), \dots , f(x_n)]</math> are restricted to lie in measurable subsets of <math>Y_n</math>. 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 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.
Line 48: Line 48:
   
 
The [[Kolmogorov continuity theorem]] guarantees that processes that satisfy certain constraints on the [[moment (mathematics)|moments]] of their increments are continuous.
 
The [[Kolmogorov continuity theorem]] guarantees that processes that satisfy certain constraints on the [[moment (mathematics)|moments]] of their increments are continuous.
  +
  +
== Examples and special cases == <!-- this part is still a bit of a mess -->
  +
  +
=== Time ===
  +
  +
A notable special case is where the time is a discrete set, for example the nonnegative integers {0, 1, 2, 3, ...}. Another important special case is <math>T = \mathbb{R}</math>.
  +
  +
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 T = {1, ..., ''n''}.
  +
  +
=== Examples ===
  +
  +
The paradigm of 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 [[wiktionary:Homogeneous|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]].
  +
  +
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 {'''X'''<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>, where a sample sequence is
  +
{'''X'''(ω)<sub>''i''</sub>}<sub>''i'' ∈ '''N'''</sub>.
  +
  +
# What is the [[probability]] that each sample sequence is [[bounded function|bounded]]?
  +
# What is the probability that each sample sequence is [[monotonic]]?
  +
# What is the probability that each sample sequence has a [[Limit of a sequence|limit]] as the index approaches ∞?
  +
# What is the probability that the [[series (mathematics)|series]] obtained from a sample sequence from <math>f(i)</math> [[Convergence (series)|converges]]?
  +
# What is the probability [[probability distribution|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)<sub>''t''</sub>}<sub>''t '' ∈ ''I''</sub>
  +
# What is the probability that it is bounded/[[integrable]]/[[continuous function|continuous]]/[[differentiable]]...?
  +
# What is the probability that it has a limit at ∞
  +
# What is the probability distribution of the integral?
   
 
==See also==
 
==See also==
 
* [[List of stochastic processes topics]]
 
* [[List of stochastic processes topics]]
* [http://www.hanoivn.net Stochastic Processes Exercises and Solutions]
+
* [[Gillespie algorithm]]
  +
* [[Markov Chain]]
  +
* [[Stochastic calculus]]
  +
* [[Dynamics of Markovian Particles|DMP]]<!-- points to disambiguation page -->
  +
* [[Covariance function]]
  +
* [[Entropy rate]] for a stochastic process
  +
* [[Stationary process]]
  +
* [[John Horton Conway]]
  +
  +
== Notes ==
  +
<references/>
   
 
==References==
 
==References==
{{cite book | author=Papoulis, Athanasios & Pillai, S. Unnikrishna | title=Probability, Random Variables and Stochastic Processes| publisher=McGraw-Hill Science/Engineering/Math | year=2001 | editor= | id=ISBN 0-07-281725-9}}
+
<div class="references-small">
  +
#{{cite book | author=Papoulis, Athanasios & Pillai, S. Unnikrishna | title=Probability, Random Variables and Stochastic Processes| publisher=McGraw-Hill Science/Engineering/Math | year=2001 | editor= | isbn=0-07-281725-9}}
  +
#{{cite web | title=Lecture notes in ''Advanced probability theory'' | author=[[Boris Tsirelson]] | url=http://www.webcitation.org/5cfvVZ4Kd}}
  +
#{{cite book | author=J. L. Doob | title=Stochastic Processes |
  +
publisher=Wiley | year=1953}}
  +
#{{cite web | title=An Exploration of Random Processes for Engineers | work=Free e-book | url=http://www.ifp.uiuc.edu/~hajek/Papers/randomprocesses.html | month=July | year=2006}}
   
 
<!-- Tips for referencing:
 
<!-- Tips for referencing:
Line 63: Line 89:
   
 
For Books, use:
 
For Books, use:
{{cite book | author=Lincoln, Abraham; Grant, U. S.; & Davis, Jefferson | title=Resolving Family Differences Peacefully | location=Gettysburg | publisher=Printing Press | year=1861 | editor=Stephen A. Douglas | id=ISBN 0-12-345678-9}}
+
{{cite book | author=Lincoln, Abraham; Grant, U. S.; & Davis, Jefferson | title=Resolving Family Differences Peacefully | location=Gettysburg | publisher=Printing Press | year=1861 | editor=Stephen A. Douglas | isbn=0-12-345678-9}}
   
   
 
For other sources, see: [[WP:CITET]]
 
For other sources, see: [[WP:CITET]]
 
-->
 
-->
{{unreferenced}}
+
</div>
  +
  +
==External links==
  +
* [http://www.youtube.com/watch?v=AUSKTk9ENzg An 8 foot tall Probability Machine (named Sir Francis) comparing stock market returns to the randomness of the beans dropping through the quincunx pattern.] from Index Funds Advisors [http://www.ifa.com IFA.com]
  +
* [http://sitmo.com/eqcat/1 Stochastic Processes used in Quantitative Finance], sitmo.com
  +
* [http://www.goldsim.com/Content.asp?PageID=455 Addressing Risk and Uncertainty]
   
 
[[Category:Stochastic processes|*]]
 
[[Category:Stochastic processes|*]]
  +
[[Category:Statistical models]]
  +
[[Category:Statistical data types]]
   
  +
<!--
 
[[ar:عملية عشوائية]]
 
[[ar:عملية عشوائية]]
  +
[[ca:Procés estocàstic]]
  +
[[cs:Náhodný proces]]
 
[[de:Stochastischer Prozess]]
 
[[de:Stochastischer Prozess]]
 
[[es:Proceso estocástico]]
 
[[es:Proceso estocástico]]
Line 80: Line 111:
 
[[gl:Proceso estocástico]]
 
[[gl:Proceso estocástico]]
 
[[it:Processo stocastico]]
 
[[it:Processo stocastico]]
  +
[[he:תהליך סטוכסטי]]
 
[[nl:Stochastisch proces]]
 
[[nl:Stochastisch proces]]
 
[[ja:確率過程]]
 
[[ja:確率過程]]
  +
[[no:Stokastisk prosess]]
 
[[pl:Proces stochastyczny]]
 
[[pl:Proces stochastyczny]]
[[pt:Estocástico]]
+
[[pt:Processo estocástico]]
[[ro:Proces stocastic]]
+
[[ro:Proces stochastic]]
 
[[ru:Случайный процесс]]
 
[[ru:Случайный процесс]]
 
[[su:Prosés stokastik]]
 
[[su:Prosés stokastik]]
 
[[fi:Stokastinen prosessi]]
 
[[fi:Stokastinen prosessi]]
  +
[[sv:Stokastisk process]]
  +
[[uk:Випадковий процес]]
 
[[vi:Quá trình ngẫu nhiên]]
 
[[vi:Quá trình ngẫu nhiên]]
 
[[zh:随机过程]]
 
[[zh:随机过程]]
  +
-->
 
{{enWP|Stochastic process}}
 
{{enWP|Stochastic process}}

Latest revision as of 11:00, March 21, 2010

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory


In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process (or deterministic system). Instead of dealing with only one possible reality of how the process might evolve under time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths may be more probable and others less.

In the simplest possible case (discrete time), a stochastic process amounts to a sequence of random variables known as a time series (for example, see Markov chain). Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as functions of one or several deterministic arguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions. Random variables corresponding to various times (or points, in the case of random fields) may be completely different. The main requirement is that these different random quantities all have the same type.[1] Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations.

Familiar examples of processes modeled as stochastic time series include signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature. Examples of random fields include static images,

Formal definition and basic properties Edit

Definition Edit

Given a probability space (\Omega, \mathcal{F}, P), a stochastic process (or random process) with state space X is a collection of X-valued random variables indexed by a set T ("time"). That is, a stochastic process F is a collection

 \{ F_t : t \in T \}

where each F_t is an X-valued random variable.

A modification G of the process F is a stochastic process on the same state space, with the same parameter set T such that

 P ( F_t = G_t) =1 \qquad \forall t \in T.

A modification is indistinguishable if

 P ( \forall t \in T \, F_t = G_t) =1 .

Finite-dimensional distributions Edit

Let F be an X-valued stochastic process. For every finite subset T' \subseteq T, we may write T'=\{ t_1, \ldots, t_k \}, where k=\left|T'\right| and the restriction F|_{T'}=(F_{t_1}, F_{t_2},\ldots, F_{t_k}) is a random variable taking values in X^k. The distribution \mathbb{P}_{T'}= \mathbb{P} (F|_{T'})^{-1} of this random variable is a probability measure on X^k. Such random variables are called the finite-dimensional distributions of F.

Under suitable topological restrictions, a suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the next section).

Construction Edit

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.

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.

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.

Kolmogorov extension Edit

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 joint 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}). Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the full-blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a Wiener process (in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. When this condition is expressed in terms of probability densities, the result is called the Chapman–Kolmogorov equation.

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.

Separability, or what the Kolmogorov extension does not provide Edit

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.

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates [f(x_1), \dots , f(x_n)] are restricted to lie in measurable subsets of Y_n. 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.

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:

  1. boundedness
  2. continuity
  3. differentiability

all require knowledge of uncountably many values of the function.

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(x_i)\} whose values determine the whole random function f.

The Kolmogorov continuity theorem guarantees that processes that satisfy certain constraints on the moments of their increments are continuous.

Examples and special cases Edit

Time Edit

A notable special case is where the time is a discrete set, for example the nonnegative integers {0, 1, 2, 3, ...}. Another important special case is T = \mathbb{R}.

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 T = {1, ..., n}.

Examples Edit

The paradigm of 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.

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.

  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}t I

  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?

See alsoEdit

Notes Edit

  1. Mathematically speaking, the type refers to the codomain of the function.

ReferencesEdit

  1. Papoulis, Athanasios & Pillai, S. Unnikrishna (2001). Probability, Random Variables and Stochastic Processes, McGraw-Hill Science/Engineering/Math.
  2. Boris Tsirelson. Lecture notes in Advanced probability theory.
  3. J. L. Doob (1953). Stochastic Processes, Wiley.
  4. (2006). An Exploration of Random Processes for Engineers. Free e-book.

External linksEdit

This page uses Creative Commons Licensed content from Wikipedia (view authors).

Around Wikia's network

Random Wiki