### Stationary Stochastic Process

A stationary 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*.

## Table of Contents

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 the 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 values 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 a non-stationary time series.

### Purely Random/ White Noise Process

A **stochastic process** having zero mean and constant variance ($\sigma^2$) and serially uncorrelated is called a 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.

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