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WeakStationarity
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WeakStationarity

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Check if a process is weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-o4er3p
Out[1]=1

Check if an autoregressive time series is weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-pcbton
Out[1]=1

Generate conditions for a time series to be weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-du0tfy
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Check if an ARProcess is weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-kzwhbe
Out[1]=1

Check if the mean function is constant in time:

In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-jl4zku
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0c0r5fg8x6giq-xtd3za
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0c0r5fg8x6giq-l56zl3
Out[4]=4

Compare covariance functions of stationary and nonstationary OrnsteinUhlenbeckProcess:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-p4m7vj
In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-dz0i91
Out[2]=2

Visualize conditions for an ARProcess to be weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-mn0j0h
Out[1]=1

For three parameters:

In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-ss40mj
Out[2]=2

Find a weakly stationary ARProcess:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-flczwd
Out[1]=1

Check:

In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-besqgr
Out[2]=2

Some processes known to be non-weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-7q866f
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-o9s6ed
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c0r5fg8x6giq-zks6mq
Out[3]=3

Some known weakly stationary processes:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-g0wriy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-l4q1di
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c0r5fg8x6giq-w77yex
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0c0r5fg8x6giq-k5eabv
Out[4]=4

Properties & Relations  (4)Properties of the function, and connections to other functions

Every MAProcess without fixed initial conditions is weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-zixm7u
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-7qua1u
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0r5fg8x6giq-hjqai1
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-zjyi97
Out[1]=1

ARIMAProcess may be weakly stationary:

In[1]:=1
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https://wolfram.com/xid/0c0r5fg8x6giq-eq5fp1
Out[1]=1
Wolfram Research (2012), WeakStationarity, Wolfram Language function, https://reference.wolfram.com/language/ref/WeakStationarity.html.
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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.

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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.

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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

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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: 25-March-2025 ]}

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@misc{reference.wolfram_2025_weakstationarity, author="Wolfram Research", title="{WeakStationarity}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WeakStationarity.html}", note=[Accessed: 25-March-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: 25-March-2025 ]}

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@online{reference.wolfram_2025_weakstationarity, organization={Wolfram Research}, title={WeakStationarity}, year={2012}, url={https://reference.wolfram.com/language/ref/WeakStationarity.html}, note=[Accessed: 25-March-2025 ]}