# Definition:Strictly Stationary Stochastic Process

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## Definition

Let $S$ be a stochastic process giving rise to a time series $T$.

$S$ is a **strictly stationary stochastic process** if its properties are unaffected by a change of origin of $T$.

## Examples

### Joint Probability Mass Function

It is necessary that:

- The joint probability mass function of any set of $m$ successive values $\set {z_1, z_2, \dotsb, z_m}$

is the same as:

- the joint probability mass function of any other set of $m$ successive values $\set {z_{1 + k}, z_{2 + k}, \dotsb, z_{m + k} }$

for an arbitrary lag $k$.

### Constant Mean Level

It is necessary that $T$ has a constant mean level, so that:

- The sample mean over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$

is the same as:

- the sample mean over any other set of $N$ successive values $\set {z_{1 + k}, z_{2 + k}, \dotsb, z_{N + k} }$

for an arbitrary lag $k$.

### Autocovariance

It is necessary that:

- The autocovariance between every two observations $z_t, z_{t + k}$ separated by a given lag $k$

is the same as:

- the autocovariance between every other two observations $z_{t + m}, z_{t + m + k}$separated by a given lag $k$

### Autocorrelation

It is necessary that:

- The autocorrelation between every two observations $z_t, z_{t + k}$ separated by a given lag $k$

is the same as:

- the autocorrelation between every other two observations $z_{t + m}, z_{t + m + k}$ separated by a given lag $k$

## Sources

- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel:
*Time Series Analysis: Forecasting and Control*(3rd ed.) ... (previous) ... (next):

- Part $\text {I}$: Stochastic Models and their Forecasting:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2.1.2$ Stationary Stochastic Processes

- $2.1$ Autocorrelation Properties of Stationary Models:

- $2$: Autocorrelation Function and Spectrum of Stationary Processes:

- Part $\text {I}$: Stochastic Models and their Forecasting: