Prosés stokastik: Béda antarrépisi

'''prosés stokastik''' nyaéta [[Fungsi (matematik)|fungsi]] [[random|acak]]. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a [[deret waktu]] in applications) or a region of space (a stochastic process being called a [[random field]]). Familiar examples of time series include [[stock market]] and [[exchange rate]] fluctuations, signals such as speech, audio and vidéo; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as [[Brownian motion]] or [[random walk]]s. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogenéous material.

:''f''_''i'' : ''W'' →→ '''R''',

:''f'' : ''D'' →→ '''R'''

For our first [[infinite]] example, take the domain to be '''N''', the [[natural numbers]], and our range to be '''R''', the [[real numbers]]. Then, a function ''f'' : '''N''' →→ '''R''' is a [[sequence]] of réal [[number]]s, and a stochastic process with domain '''N''' and range '''R''' is a random sequence. The following questions arise:

# How is a [[random sequence]] specified?

# How do we find the answers to typical questions about sequences, such as

## what is the [[probability]] that ''f'' is [[bounded]]?

## what is the probability that ''f''(''i'') has a [[limit]] as ''i''→∞→∞?

Another important class of examples is when the domain is not a [[discrete space]] such as the natural numbers, but a [[continuous space]] such as the [[unit interval]] [0,1], the positive réal numbers [0,∞∞) or the entire [[real line]], '''R'''. In this case, we have a different set of questions that we might want to answer:

# How is a random function specified?

# How do we find the answers to typical questions about functions, such as

## what is the probability that ''f'' is bounded/[[integrable]]/[[continuous]]/[[differentiable]]...?

## what is the probability that ''f''(''x'') has a limit as ''x''→∞→∞ ?

* [[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-homogenéous.

* [[process with independent increments|Processes with independent increments]]: processes where the domain is at léast partially ordered and, if ''x''₁ <...< ''x_n'', all the variables ''f''(''x''_k+1) &minus;− ''f''(''x_k'') are independent. [[Markov chain]]s are a special case.

* [[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 ''S''. They can be modélled 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 process]]es: processes where all linéar 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]]s—processes with constraints on the expectation

* [[Galton-Watson process]]es

* [[Elevator paradox]]

* [[Branching process]]es

* [[Bernoulli process]]es

The Kolmogorov extension proceeds along the following lines: assuming that a [[probability measure]] on the space of all functions ''f'' : ''X'' &rarr;→ ''Y'' exists, then it can be used to specify the probability distribution of finite-dimensional random variables [''f''(''x''₁),...,''f''(''x_n'')]. Now, from this ''n''-dimensional probability distribution we can deduce an ''(n-1)''-dimensional [[marginal probability distribution]] for [''f''(''x''₁),...,''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 density function|probability densities]], the result is called the [[Chapman-Kolmogorov equation]].

# [[bounded]]ness

# [[continuity]]

# [[differentiability]]

# ''E''(''a''*''a'')&ge;0≥0 for all random variables ''a'';

* [Box and Jenkins]'' Time Series Analysis Forecasting And Control'', &nbsp;Géorge Box, Gwilym Jenkins, &nbsp;Holden-Day (1976) &nbsp;ISBN 0-8162-1104-3

* [Doob]'' Stochastic Processes, ''J. L Doob, John Wiley &amp; Sons (1953) Library of Congress Catalog Number: 52-11857

* [Gardiner] ''Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences'', &nbsp;Second edition, &nbsp;C.W. Gardiner, Springer Verlag (1985) ISBN 3-540-15607-0

* [Iyanaga and Kawada] ''Encyclopedic Dictionary Of Mathematics ''Volume II, edited by Shokichi Iyanaga and Yukiyosi Kawada, translated by Kenneth May, &nbsp;MIT Press (1980) &nbsp;ISBN 0-262-59010-7

* [Karlin and Taylor]'' &nbsp;A First Course In Stochastic Processes'', &nbsp;second edition,&nbsp; Samuel Karlin, Howard Taylor, Academic Press (1975) &nbsp;ISBN 0-12-398552-8

* [Neftci]'' An Introduction To The Mathematics Of Financial Derivatives'', Salih Neftci, Academic Press (1996) &nbsp;ISBN 0-12-515390-2

* [Parzen]''Stochastic Processes,'' Emmanuel Parzen, &nbsp;Holden-Day (San Francisco 1962) &nbsp;ISBN 0-8162-6664-6

* [Vanmarcke] Random Fields: Analysis and Synthesis, Erik VanMarcke, MIT Press (1983) &nbsp;ISBN 0-262-22026-1 &nbsp; <a web edition is available.</a><br>&nbsp;