Let Z_t, t = ..., -2, -1, 0, 1, 2, ..., be a strictly stationary white noise process and let a, b, c be constants. Which of the following processes are stationary? For each stationary process specify whether it is strongly or weakly stationary:
a). X_t = a + b Z_t + c Z_(t-1)
b). X_t = a + b Z_0
c). X_t = cos(ct) Z_1 + sin(ct) Z_2
d). X_t = cos(ct) Z_0
e). X_t = cos(ct) Z_t + sin(ct) Z_(t-1)
f). X_t = Z_t Z_(t-1)
I understand that in order to show that a process is stationary (and hence weakly stationary) I need to show that the mean is a constant and that the autocovariance function does not depend on t. However, how can I show that a stationary process is strongly stationary? I think that it is necessary to show that it is Gaussian, but I do not know how it is possible to show that a stationary process is Gaussian in a simple yet acceptable way. Please pick any process from the list above and show me how it is possible to prove that it is strongly stationary. I would appreciate your help and hints!
Thank you for the reply. Can you please explain how to prove that a stationary process is strictly stationary or not strictly stationary (aka strongly stationary)? As of your comments about the given processes being stationary (i.e., weakly stationary), my answers differ from yours, so I would like to summarize my answers below to see which answers are right.
Let Z_t, t = ..., -2, -1, 0, 1, 2, ..., be a strictly stationary white noise process and let a, b, c be constants.
a). X_t = a + b Z_t + c Z_(t-1)
Z_t has zero mean and variance sigma^2 and autocovariance function y_Z (h) = sigma^2 if h = 0, and 0 if h does not equal 0. Since the Z_t are strictly stationary, then they are independent and normally distributed.
The mean E(X_t) = a is a constant, and the autocovariance function is y_X (h) = Cov(X_t+h, X_t) =
= (b^2 + c^2) sigma^2 if h = 0;
= bc sigma^2 if h = -1 or h = 1;
= 0 if the absolute value of h is greater than 1.
The autocovariance function does not depend on t. Hence, X_t is stationary.
b). X_t = a + b Z_0
The mean E(X_t) = a is a constant, and the autocovariance function y_X (h) = Cov(X_t+h, X_t) = b^2 sigma^2 does not depend on t, so X_t is stationary.
c). X_t = cos(ct) Z_1 + sin(ct) Z_2
The mean E(X_t) = 0 is a constant, and the autocovariance function y_X (h) = Cov(X_t+h, X_t) = sigma^2 cos(ch) does not depend on t, so X_t is stationary.
d). X_t = cos(ct) Z_0
The mean E(X_t) = 0 is a constant, and the autocovariance function y_X (h) = Cov(X_t+h, X_t) = cos^2 (ct) sigma^2 depends on t unless c = 2 k pi, so X_t is NOT stationary unless c = 2 k pi.
e). X_t = cos(ct) Z_t + sin(ct) Z_(t-1)
The mean E(X_t) = 0 is a constant, and the autocovariance function is y_X (h) = Cov(X_t+h, X_t) =
= sigma^2 cos(ch) if h = 0;
= cos(ct+ch) sin(ct) sigma^2 if h = -1;
= sin(ct+ch) cos(ct) sigma^2 if h = 1;
= if the absolute value of h is greater than 1.
It depends on t unless c = 2 k pi, so X_t is NOT stationary unless c = 2 k pi.
f). X_t = Z_t Z_(t-1)
The mean E(X_t) = 0 is a constant, and the autocovariance function y_X (h) = Cov(X_t+h, X_t) = 0 does not depend on t (since the covariance of Z_t and Z_t-1 is zero due to their independence), so X_t is stationary.
OK, thank you anyway. I really hope that someone here knows the concept of strict stationarity and its examples since my textbooks do not provide good examples of how to prove that a process is strictly stationary or not strictly stationary. I also tried Internet, but in vain. If you know some website that might help me solve my problems, I would appreciate that.