  In this article we introduce and study oscillating Gaussian processes defined
by $X_t = \alpha_+ Y_t {\bf 1}_{Y_t >0} + \alpha_- Y_t{\bf 1}_{Y_t<0}$, where
$\alpha_+,\alpha_->0$ are free parameters and $Y$ is either stationary or
self-similar Gaussian process. We study the basic properties of $X$ and we
consider estimation of the model parameters. In particular, we show that the
moment estimators converge in $L^p$ and are, when suitably normalised,
asymptotically normal.
