# Computing the multivariate CDF

I was reading these slides, in the last slide it is stated that the process $$\{Z_t : t=1,2,\cdots\}$$ is not strongly stationary. It is defined as: $$Z_t = \cos(tw)$$

Where $$w$$ is a random variable uniformly distributed in the interval $$(0,2π)$$. A process is strongly stationary if for all the sets of index $$t_i$$ verifies: $$F_{t_{1}+k, t_{2}+k, \cdots, t_{s}+k}\left(b_{1}, b_{2}, \cdots, b_{s}\right)=F_{t_{1}, t_{2}, \cdots, t_{s}}\left(b_{1}, b_{2}, \cdots, b_{s}\right)$$

I was wondering how to prove the claim. How can I obtain the multivariate CDF of the process $$Z_t$$?

All you need is one example where that equality fails. For example, $$P(Z_1 \le 0 \cap Z_3 \le 0 )$$ is equal to (it is easiest to draw a picture of the respective cosine curves to see why this is so) $$P( \{ w \in [\pi/2, 3 \pi/2] \} \cap \{ w \in [\pi/6, 3 \pi/6]\cup [ 2\pi/3 + \pi/6, 2\pi/3 + 3\pi/6] \cup [ 4\pi/3 + \pi/6, 4\pi/3 + 3\pi/6]\})=P(w \in [ 2\pi/3 + \pi/6, 2\pi/3 + 3\pi/6] )= 1/6$$
but, from a similar calculation, $$P(Z_2 \le 0 \cap Z_4 \le 0 )=1/4$$