I am studying random process. The concept is not familiar to me. In the book, it says
"signal $\cos(wt+\theta)$ where $\theta$ is uniform over $(0, 2\pi)$ is strict sense stationary".
But it gives no explanation or proof. I know how to prove that sinusoidal signal is wide sense stationary. However, I could not prove it and could not find the proof of strict sense stationary on the internet. All the proofs are about wide sense stationary. I don't know why. Strict sense stationary is not widely used? Could you help me?
Let $X(t)=A\cos(\omega t+\Theta)$, where $A$, $\omega$ are constants and $\Theta\sim\text{Unif}[0,2\pi)$. For any $\tau$, we have $X(t+\tau)=A\cos(\omega t+\omega\tau+\Theta)=A\cos(\omega t+\tilde{\Theta})$, where $$ \tilde{\Theta}=(\omega\tau+\Theta)\mod 2\pi $$ with the mod operation resulting in a value in $[0,2\pi)$. Since $\Theta$ is uniform and the mod operation is a one-to-one map, we see that $\tilde{\Theta}$ and $\Theta$ are identically distributed for any $\tau$.
Now, $[X(t_1+\tau),X(t_2+\tau),\ldots,X(t_k+\tau)]$ simplifies to $[A\cos(\omega t_1+\tilde{\Theta}),A\cos(\omega t_2+\tilde{\Theta}),\ldots,A\cos(\omega t_k+\tilde{\Theta})]$, which (using the argument above) has the same distribution as $[A\cos(\omega t_1+\Theta),A\cos(\omega t_2+\Theta),\ldots,A\cos(\omega t_k+\Theta)]$ for any $k$, $t_1,t_2,\ldots,t_k$ and $\tau$. This proves strict sense stationarity.