Proof of strictly stationary process

In summary, the question is whether process Y_n=1/2*X_n+1/4*X_{n-1}+1/8*X_{n-2} is strictly stationary. The answer is yes, as the Y's will be correlated but have identical distributions due to the identical distributions of the X's. This shows that the process is strictly stationary.
  • #1
trenekas
61
0
Hi all. I need to prove or disprove if process [itex]Y_n=1/2*X_n+1/4*X_{n-1}+1/8*X_{n-2}[/itex] are stricly stationary. [itex]X_n,n\in R[/itex] i.i.d.
So almost i have the answer. But don't know if it is correct or not. I have a question of situation when [itex]\Gamma_Y(t,s)[/itex] and |t-s|≤2 for example:
[itex]Y_{10}=1/2*X_{10}+1/4*X_{9}+1/8*X_{8}[/itex]
[itex]Y_8=1/2*X_8+1/4*X_{7}+1/8*X_{6}[/itex]
There are [itex]x_8[/itex] in both but situations and not sure if [itex]Y_{10}[/itex] and [itex]Y_{8}[/itex] are i.d.
Hope you understand my problem.
Maybe its a little stupid question but I'm just started to learn random processes. :)
Thanks for help.
 
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  • #2
The Y's will be correlated as you noticed. The correlation depends only on the parameter difference.
However, the distributions will be identical, since the X's have identical distributions.

Put these together and you get strictly stationary.
 

What is a strictly stationary process?

A strictly stationary process is a stochastic process where the joint distribution of any finite subset of random variables remains unchanged over time. This means that the statistical properties of the process, such as mean and variance, do not change over time.

What is the difference between strict stationarity and weak stationarity?

Strict stationarity requires the joint distribution of all subsets of random variables to remain unchanged over time, while weak stationarity only requires the first and second moments of the process to be constant over time.

How is the autocovariance function used to determine strict stationarity?

The autocovariance function measures the linear dependency between two observations at different time points. For a process to be strictly stationary, the autocovariance function should only depend on the time lag between observations, not the specific time points.

Why is strict stationarity important in time series analysis?

Strict stationarity allows for the use of many powerful statistical tools, such as the autocorrelation function and spectral density, to analyze and model time series data. It also simplifies the process of forecasting future values of the process.

How can strict stationarity be tested for in a time series?

There are several statistical tests, such as the Ljung-Box test and the Box-Pierce test, that can be used to determine if a time series process is strictly stationary. These tests compare the autocorrelation function of the process to the expected values under the null hypothesis of strict stationarity.

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