Let me start out and say I am very new to statistics and am trying to do some self-learning. Some definitions that may seem trivial, I am still learning.
Given some random process:
$$X_t = cos(t\omega_k + \theta_k)$$
If $w_k = 8\pi$, and $\theta_k$ is randomly sampled from $[0,\pi]$, is the process first order stationary and mean ergodic?
- Finding out if it is stationary
I am not having the easiest time with the definition I have found on Wikipedia:
Let $F_X(x_{t1+\tau},...x_{t_n+\tau})$ represent the cumulative distribution function for the unconditional, if $F_X(x_{t1+\tau},...x_{t_n+\tau}) = F_X(x_{t1},...x_{t_n})$, the process is stationary.
How do we find the cumulative distribution function of a process that has several variables like this?
$$\int_0^\pi cos(8\pi t + \theta_k) d\theta$$
I assume once I have this I just need to see if the CDF produces different results for different $t$.
While I can see easily how we could disprove stationarity, what would we do in order to prove it. I.e. we need to somehow show for all values of $t$ the values are the same above.
- Finding out if it is mean ergodic
For this we would need to see if the mean ($E[X(t)]$) remains constant for all t.
In this case, would the mean calculation be done by:
$$\int_0^\pi \theta\ cos(8\pi t + \theta_k) d\theta$$
?
For this part I am not actually sure where to go with it
Thank you for your time