# Example and simulated path of strict(ly) stationary process

There are many question on stationary process but very few related to strict stationary process. I am just looking for an example and simulated path of strict stationary process and how strict stationary process look different from weak stationary process and white noise.

I found following example of strict stationary process from Wikipedia:

$$X_t=\cos (t+Y) \quad \text{ for } t \in \mathbb{R},$$ where $Y$ have a uniform distribution on $(0,2π]$. I can not understand how the above process is strictly stationary? Alternative simple example of strictly stationary process would also be appreciated.

[EDIT]Simulated path for above process:

with acf and pacf as follows :

• Try plotting histograms of many draws for different values of $t$; you will see that the uniform dist'n for $Y$ renders the value of $t$ meaningless when run through the cosine function. Also, there's no hard and fast rule for how strictly stationary processes "look different" from weakly stationary processes. – jbowman Jan 12 '16 at 19:20
• @jbowman I have added simulated path and ACF and PACF for strict stationary process. The process is just like white noise process. So, then what the difference between white noise and strict white noise process ? – Neeraj Jan 12 '16 at 19:48
• A white noise process is strictly stationary. For an example of a weakly, but not strictly, stationary process, consider one with $x_t$ distributed standard normal when $t$ is even and equal to either -1 or 1, each with probability 0.5, when $t$ is odd. The mean and covariance function are independent of $t$, but the joint distribution changes if you step from an "origin" of $t_0$ even to one of $t_0$ odd. – jbowman Jan 12 '16 at 20:31