Oscillating Gaussian Processes

Abstract

In this article we introduce and study oscillating Gaussian processes defined by $X_t={\alpha }_{+}{Y}_t{1}_{Y_t>0}+{\alpha }_{-}{Y}_t{1}_{Y_t<0}$, 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.