# 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_2024_weakstationarity, author="Wolfram Research", title="{WeakStationarity}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WeakStationarity.html}", note=[Accessed: 13-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_weakstationarity, organization={Wolfram Research}, title={WeakStationarity}, year={2012}, url={https://reference.wolfram.com/language/ref/WeakStationarity.html}, note=[Accessed: 13-September-2024 ]}