Is $Z_t=A+\sin(2\pi t+\theta)$ stationary?

Question

If $Z_t=A+\sin(2\pi t+\theta)$ is a discrete-time time series with $A\in\mathbb{R}$ and $\theta\sim U[0,2\pi]$. Is $Z_t$ stationary? If $A$ is a random variable with mean $0$ and variance $1$ and $\theta\in\mathbb{R}$, is $Z_t$ stationary?

Using the $\sin$ properties $$\sin(2\pi t+\theta)=\sin(2\pi t)\cos(\theta)+\sin(\theta)\cos(2\pi t)$$ Since $t$ is discrete then $$\sin(2\pi t+\theta)=A\sin(\theta)$$

and it is clearly stationary. The same is true in the second case when $A$ is a random variable.

Is there some trick here?

Answer 1

If $\theta$ is uniform on $(0,2\pi)$ and $A$ is independent of $\theta$ (in particular, if $A$ is constant), then the distribution of $Z_t$ does not depend on the real number $t$.

Reason: For every $t$, $2\pi t+\theta$ is uniform on the interval $(2\pi t,2\pi t+2\pi)$ hence $\sin(2\pi t+\theta)$ is distributed as $\sin(\theta)$. Furthermore, $\sin(2\pi t+\theta)$ and $\sin(\theta)$ are both independent of $A$ hence $Z_t=A+\sin(2\pi t+\theta)$ and $Z_0=A+\sin(\theta)$ follow the same distribution.

The same approach shows that the distribution of every $(Z_{t+t_1},Z_{t+t_2},\ldots,Z_{t+t_1})$ does not depend on $t$.

Answer 2

If $t$ is an integer then the value of $Z_t$ does not depend on the time $t$. Consequently, the joint distribution of $Z_{t_1}, Z_{t_2}, \cdots, Z_{t_n}$ for $n \ge 1$ does not depend on the $t_i$'s. Consequently, $Z_t$ is a stationary process.