The Bussgang theorem states that if a stationary Gaussian process $X(t)$ with covariance function $R(\tau)$ is passing any non-linear function, affecting only the amplitude, $g(x)$, then the covariance of $y(t)=g(X(t))$ is $CR(\tau)$, for some C.

Now, let the process be instead cyclo-stationary with covariance $R(\tau_1,\tau_2)=R(\tau_1-\tau_2,\tau_2)=R(\tau_1-\tau_2,\tau_2+T)$, where $T$ is the period. Is it true that the covariance of $Y(t)=g(X(t))$ is $CR(\tau_1,\tau_2)$, possibly with another constant $C$?