Given $X_{t} = sin(pt + U_{t})$ where $U_{t}$ is uniformly distributed over $[0,2\pi]$. Check the stationarity of this process.
It is given as non-stationary in the answer sheet. I was not able to understand their explanation. Here is what I have done.
For a process to be stationary, I need to show the following two results:
- $E[X_{t}] =$ some constant.
- $Cov(X_{t}, X_{t+s}) = $ should depend solely on s.
$E[X_{t}] = E[sin(pt + U_{t})] = \int_{0}^{2\pi}sin(pt + U_{t})\frac{1}{2\pi}du = 0$
Now,
$ Cov(X_{t},X_{t+s}) = \int_{0}^{2\pi}sin(pt + U_{t})sin(p(t+s) + U_{t})\frac{1}{2\pi}du = \pi cos(ps)$
From this result, It is clear that the covariance depends on s and also mean is 0 which is constant. I don't understand the book's explanation.
I can show my complete steps but I have purposely skipped the middle steps because that would mean lot of typing in latex. Here, to solve the covariance function I have used the following formula:
$ Sin(A) * Sin(B) = \frac{1}{2} * [cos(A-B) - cos(A+B)] $
I just want to understand whether What I have done is correct or not?
In the answer sheet, they have just written following explanation which I was not able to understand:
This is not purely indeterministic, and is not therefore a stationary time series in the sense defined in the text.
