**Stationary Stochastic Process**

**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 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)

*(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 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.

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