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WeakStationarity[proc]

gives conditions for the process proc to be weakly stationary.

Details
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Basic Examples  
Scope  
Properties & Relations  
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WeakStationarity

WeakStationarity[proc]

gives conditions for the process proc to be weakly stationary.

Details

  • Weakly stationary processes are also known as wide-sense stationary or covariance stationary.
  • A random process proc is weakly stationary if its mean function is independent of time, and its covariance function is independent of time translation.

Examples

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Basic Examples  (3)

Check if a process is weakly stationary:

Check if an autoregressive time series is weakly stationary:

Generate conditions for a time series to be weakly stationary:

Scope  (6)

Check if an ARProcess is weakly stationary:

Check if the mean function is constant in time:

Check if the covariance function is a function of time difference:

Compare covariance functions of stationary and nonstationary OrnsteinUhlenbeckProcess:

Visualize conditions for an ARProcess to be weakly stationary:

For three parameters:

Find a weakly stationary ARProcess:

Check:

Some processes known to be non-weakly stationary:

Some known weakly stationary processes:

Properties & Relations  (4)

Every MAProcess without fixed initial conditions is weakly stationary:

Time series processes with fixed initial conditions are not weakly stationary:

The conditions for an ARMAProcess to be weakly stationary depend only on the autoregressive parameters:

ARIMAProcess may be weakly stationary:

Wolfram Research (2012), WeakStationarity, Wolfram Language function, https://reference.wolfram.com/language/ref/WeakStationarity.html.

Text

Wolfram Research (2012), WeakStationarity, Wolfram Language function, https://reference.wolfram.com/language/ref/WeakStationarity.html.

CMS

Wolfram Language. 2012. "WeakStationarity." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeakStationarity.html.

APA

Wolfram Language. (2012). WeakStationarity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeakStationarity.html

BibTeX

@misc{reference.wolfram_2025_weakstationarity, author="Wolfram Research", title="{WeakStationarity}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WeakStationarity.html}", note=[Accessed: 11-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_weakstationarity, organization={Wolfram Research}, title={WeakStationarity}, year={2012}, url={https://reference.wolfram.com/language/ref/WeakStationarity.html}, note=[Accessed: 11-August-2025]}