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: