Determine stationarity of time series containing sin of white noise

Question

Could someone help me determine the stationarity of the the following time series Y? $ Z_t $ represents white noise with variance $ \sigma^2 $.

$ Y_t = \sin(Z_t) + Z^2_t - Z_{t-1}$

I have tried calculating $ E(Y_t) $ and $ E(Y_tY_{t-1}) $ to determine if they rely on $ t $ but I got stuck calculating $ E(\sin(Z_t)) $

Answer 1

Actually, you do not need an explicit formula for $E(\sin(Z_t))$; the fact that $Z_t$ has the same distribution as $Z_0$ shows that $E(\sin(Z_t))=E(\sin(Z_0))$.

Actually, we can use the following:

Let $\left(\xi_i\right)_{i\in\mathbb Z}$ be an i.i.d. sequence and let $f\colon \mathbb R^2\to\mathbb R$ be a measurable function. Then the sequence $\left(X_n\right)_{n\geqslant 1}$ defined by $X_n=f\left(\xi_i,\xi_{i-1}\right)$ is strictly stationary. In particular, if $X_1$ is square integrable, then $\left(X_n\right)_{n\geqslant 1}$ is stationary in the weak sense.