Stationary process

stationarynon-stationarystationaritywide-sense stationarystationary stochastic processwide sense stationarycovariance-stationaryjointly wide-sense stationaryNon-stationaritynon-stationary process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.wikipedia
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Cyclostationary process

cyclostationarityCyclostationarycyclostationary noise
An important type of non-stationary process that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.
A cyclostationary process can be viewed as multiple interleaved stationary processes.

Time series

time series analysistime-seriestime-series analysis
Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary.
The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model).

Unit root

difference stationarynon-stationary
The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend.
Such a process is non-stationary but does not always have a trend.

Trend stationary

trendtrend-stationaritytrend-stationary
In the latter case of a deterministic trend, the process is called a trend stationary process, and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.
In the statistical analysis of time series, a stochastic process is trend stationary if an underlying trend (function solely of time) can be removed, leaving a stationary process.

Autoregressive–moving-average model

ARMAautoregressive moving average modelautoregressive moving average
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).

Autoregressive model

autoregressiveautoregressionAutoregressive process
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.
Contrary to the moving-average model, the autoregressive model is not always stationary as it may contain a unit root.

Moving-average model

Moving average modelmoving averagemoving average process
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.
Contrary to the AR model, the finite MA model is always stationary.

Autocorrelation

autocorrelation functionserial correlationautocorrelated
This also implies that the autocorrelation depends only on, that is
If is a wide-sense stationary process then the mean \mu and the variance \sigma^2 are time-independent, and further the autocovariance function depends only on the lag between t_1 and t_2: the autocovariance depends only on the time-distance between the pair of values but not on their position in time.

Autocovariance

autocovariance functionautocovariance matrixmean and autocovariance
WSS random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time and that the 2nd moment is finite for all times.
If is a weakly stationary (WSS) process, then the following are true:

Bernoulli scheme

Bernoulli shiftBernoulli automorphismBernoulli measure
An example of a discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is a Bernoulli scheme.
These include all finite stationary stochastic processes, subshifts of finite type, finite Markov chains, Anosov flows, and Sinai's billiards: these are all isomorphic to Bernoulli schemes.

Wiener–Khinchin theorem

Wiener-Khinchin theoremWiener-Khintchine theoremWiener–Khinchin–Einstein theorem
In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.

Stationary ergodic process

In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity.

Bochner's theorem

Bochner theoremBochner’s theorem
By the positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure \mu on the real line such that H is isomorphic to the Hilbert subspace of L 2 generated by {e −2πiξ⋅t }.
A sequence of random variables \{ f_n \} of mean 0 is a (wide-sense) stationary time series if the covariance

Mathematics

mathematicalmathmathematician
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.

Statistics

statisticalstatistical analysisstatistician
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.

Mean

mean valueaveragepopulation mean
Consequently, parameters such as mean and variance also do not change over time.

Variance

sample variancepopulation variancevariability
Consequently, parameters such as mean and variance also do not change over time.

Mean reversion (finance)

mean reversionmean-revertingmean-reversion
In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
Formally, let be a stochastic process and let represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of at times.

Marginal distribution

marginal probabilitymarginalmarginals
Formally, let be a stochastic process and let represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of at times.

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Formally, let be a stochastic process and let represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of at times.

White noise

whitenoisestatic
White noise is the simplest example of a stationary process.

Random variable

random variablesrandom variationrandom
Let Y be any scalar random variable, and define a time-series, by

Law of large numbers

strong law of large numbersweak law of large numbersBernoulli's Golden Theorem
A law of large numbers does not apply on this case, as the limiting value of an average from a single realisation takes the random value determined by Y, rather than taking the expected value of Y.