Stationary Stochastic Process

A stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on a distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. Such a stochastic process is also known as weak stationary, covariance stationary, second-order stationary or wide sense stochastic process.

In other words, a sequence of random variables {$y_t$} is covariance stationary if there is no trend, and if the covariance does not change over time.

Strictly Stationary (Covariance Stationary)

A time series is strictly stationary if all the moments of its probability distribution are invariance over time but not for first two (mean and variance).

Let $y_t$ be a stochastic time series with

$E(y_t) = \mu $    $\Rightarrow$ Mean
$V(y_t) = E(y_t -\mu)^2=\sigma^2 $  $\Rightarrow$ Variance
$\gamma_k = E[(y_t-\mu)(y_{t+k}-\mu)]$  $\Rightarrow$ Covariance = $Cov(y_t, y_{t-k})$

$\gamma_k$ is covariance or autocovariance at lag $k$.

If $k=0$ then $Var(y_t)=\sigma^2$ i.e. $Cov(y_t)=Var(y_t)=\sigma^2$

If $k=1$ then we have covariance between two adjacent value of $y$.

If $y_t$ is to be stationary, the mean, variance and autocovariance of $y_{t+m}$ (shift or origin of $y=m$) must be the same as those of $y_t$. OR

If a time series is stationary, its mean, variance and autocovariance remain the same no matter at what point we measure them, i.e, they are time invariant.

Non-Stationary Time Series

A time series having a time-varying mean or a time-varying variance or both is called non-stationary time series.

Purely Random/ White Noise Process

A stochastic process having zero mean and a constant variance ($\sigma^2$) and serially uncorrelated is called purely random/ white noise process.

If it is independent also then such a process is called strictly white noise.

White noise denoted by $\mu_t$ as $\mu_t \sim N(0, \sigma^2)$ i.e. $\mu_t$ is independently and identically distributed as a normal distribution with zero mean and constant variance.

Stationary time series is important because if a time series is non-stationary, we can study its behaviour only for the time period under consideration. Each set of time series data will, therefore, be for a particular episode. As consequence, it is not possible to generalize it to other time periods. Therefore, for the purpose of forecasting, such (non-stochastic) time series may be of little practical value. Our interest is in stationary time series.