Oscillating Gaussian processes

Abstract

AbstractIn this article we introduce and study oscillating Gaussian processes defined by $$X_t = \alpha _+ Y_t \mathbf{1}_{Y_t >0} + \alpha _- Y_t\mathbf{1}_{Y_t<0}$$ X t = α + Y t 1 Y t > 0 + α - Y t 1 Y t < 0 , where $$\alpha _+,\alpha _->0$$ α + , α - > 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 $$L^p$$ L p and are, when suitably normalised, asymptotically normal.