Stationary process

In mathematics and statistics, a stationary process (a.k.a. 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. Consequently, parameters such as mean and variance also do not change over time. F X ( x t 1 + τ , … , x t n + τ ) = F X ( x t 1 , … , x t n ) for all  τ , t 1 , … , t n ∈ R  and for all  n ∈ N {displaystyle F_{X}(x_{t_{1}+ au },ldots ,x_{t_{n}+ au })=F_{X}(x_{t_{1}},ldots ,x_{t_{n}})quad { ext{for all }} au ,t_{1},ldots ,t_{n}in mathbb {R} { ext{ and for all }}nin mathbb {N} }     (Eq.1) F X ( x t 1 + τ , … , x t n + τ ) = F X ( x t 1 , … , x t n ) for all  τ , t 1 , … , t k ∈ R  and for all  n ∈ { 1 , … , N } {displaystyle F_{X}(x_{t_{1}+ au },ldots ,x_{t_{n}+ au })=F_{X}(x_{t_{1}},ldots ,x_{t_{n}})quad { ext{for all }} au ,t_{1},ldots ,t_{k}in mathbb {R} { ext{ and for all }}nin {1,ldots ,N}}     (Eq.2) m X ( t ) = m X ( t + τ ) for all  τ ∈ R K X X ( t 1 , t 2 ) = K X X ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R E [ | X ( t ) | 2 ] < ∞ for all  t ∈ R {displaystyle {egin{aligned}&m_{X}(t)=m_{X}(t+ au )quad &{ ext{for all }} au in mathbb {R} \&K_{XX}(t_{1},t_{2})=K_{XX}(t_{1}-t_{2},0)quad &{ ext{for all }}t_{1},t_{2}in mathbb {R} \&mathbb {E} <infty quad &{ ext{for all }}tin mathbb {R} end{aligned}}}     (Eq.3) m X ( t ) = m X ( t + τ ) for all  τ ∈ R K X X ( t 1 , t 2 ) = K X X ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R J X X ( t 1 , t 2 ) = J X X ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R E [ | X ( t ) | 2 ] < ∞ for all  t ∈ R {displaystyle {egin{aligned}&m_{X}(t)=m_{X}(t+ au )&quad { ext{for all }} au in mathbb {R} \&K_{XX}(t_{1},t_{2})=K_{XX}(t_{1}-t_{2},0)&quad { ext{for all }}t_{1},t_{2}in mathbb {R} \&J_{XX}(t_{1},t_{2})=J_{XX}(t_{1}-t_{2},0)&quad { ext{for all }}t_{1},t_{2}in mathbb {R} \&mathbb {E} <infty quad &{ ext{for all }}tin mathbb {R} end{aligned}}}     (Eq.4) F X Y ( x t 1 , … , x t m , y t 1 ′ , … , y t n ′ ) = F X Y ( x t 1 + τ , … , x t m + τ , y t 1 ′ + τ , … , y t n ′ + τ ) for all  τ , t 1 , … , t m , t 1 ′ , … , t n ′ ∈ R  and for all  m , n ∈ N {displaystyle F_{XY}(x_{t_{1}},ldots ,x_{t_{m}},y_{t_{1}^{'}},ldots ,y_{t_{n}^{'}})=F_{XY}(x_{t_{1}+ au },ldots ,x_{t_{m}+ au },y_{t_{1}^{'}+ au },ldots ,y_{t_{n}^{'}+ au })quad { ext{for all }} au ,t_{1},ldots ,t_{m},t_{1}^{'},ldots ,t_{n}^{'}in mathbb {R} { ext{ and for all }}m,nin mathbb {N} }     (Eq.5) F X Y ( x t 1 , … , x t m , y t 1 ′ , … , y t n ′ ) = F X Y ( x t 1 + τ , … , x t m + τ , y t 1 ′ + τ , … , y t n ′ + τ ) for all  τ , t 1 , … , t m , t 1 ′ , … , t n ′ ∈ R  and for all  m ∈ { 1 , … , M } , n ∈ { 1 , … , N } {displaystyle F_{XY}(x_{t_{1}},ldots ,x_{t_{m}},y_{t_{1}^{'}},ldots ,y_{t_{n}^{'}})=F_{XY}(x_{t_{1}+ au },ldots ,x_{t_{m}+ au },y_{t_{1}^{'}+ au },ldots ,y_{t_{n}^{'}+ au })quad { ext{for all }} au ,t_{1},ldots ,t_{m},t_{1}^{'},ldots ,t_{n}^{'}in mathbb {R} { ext{ and for all }}min {1,ldots ,M},nin {1,ldots ,N}}     (Eq.6) m X ( t ) = m X ( t + τ ) for all  τ ∈ R m Y ( t ) = m Y ( t + τ ) for all  τ ∈ R K X X ( t 1 , t 2 ) = K X X ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R K Y Y ( t 1 , t 2 ) = K Y Y ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R K X Y ( t 1 , t 2 ) = K X Y ( t 1 − t 2 , 0 ) for all  t 1 , t 2 ∈ R {displaystyle {egin{aligned}&m_{X}(t)=m_{X}(t+ au )&quad { ext{for all }} au in mathbb {R} \&m_{Y}(t)=m_{Y}(t+ au )&quad { ext{for all }} au in mathbb {R} \&K_{XX}(t_{1},t_{2})=K_{XX}(t_{1}-t_{2},0)&quad { ext{for all }}t_{1},t_{2}in mathbb {R} \&K_{YY}(t_{1},t_{2})=K_{YY}(t_{1}-t_{2},0)&quad { ext{for all }}t_{1},t_{2}in mathbb {R} \&K_{XY}(t_{1},t_{2})=K_{XY}(t_{1}-t_{2},0)&quad { ext{for all }}t_{1},t_{2}in mathbb {R} end{aligned}}}     (Eq.7) In mathematics and statistics, a stationary process (a.k.a. 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. Consequently, parameters such as mean and variance also do not change over time. Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become 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. In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. 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. A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. 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. For many applications strict-sense stationarity is too restrictive. Other forms of stationarity such as wide-sense stationarity or N-th order stationarity are then employed. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology). Formally, let { X t } {displaystyle left{X_{t} ight}} be a stochastic process and let F X ( x t 1 + τ , … , x t n + τ ) {displaystyle F_{X}(x_{t_{1}+ au },ldots ,x_{t_{n}+ au })} represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of { X t } {displaystyle left{X_{t} ight}} at times t 1 + τ , … , t n + τ {displaystyle t_{1}+ au ,ldots ,t_{n}+ au } . Then, { X t } {displaystyle left{X_{t} ight}} is said to be strictly stationary, strongly stationary or strict-sense stationary if:p. 155 Since τ {displaystyle au } does not affect F X ( ⋅ ) {displaystyle F_{X}(cdot )} , F X {displaystyle F_{X}} is not a function of time. White noise is the simplest example of a stationary process. 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. 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. Models with a non-trivial autoregressive component may be either stationary or non-stationary, depending on the parameter values, and important non-stationary special cases are where unit roots exist in the model. Let Y {displaystyle Y} be any scalar random variable, and define a time-series { X t } {displaystyle left{X_{t} ight}} , by