Zero-crossing rates of mixtures and products of Gaussian processes

Formulas for the expected zero-crossing rate of non-Gaussian mixtures and products of Gaussian processes are obtained. The approach we take is to first derive the expected zero-crossing rate in discrete time and then obtain the rate in continuous time by an appropriate limiting argument. The processes considered, which are non-Gaussian but derived from Gaussian processes, serve to illustrate the variability of the zero-crossing rate in terms of the normalized autocorrelation function p(t) of the process. For Gaussian processes, Rice\´s formula gives the expected zero-crossing rate in continuous time as 1/π√(-ρ"(0)). We show processes exist with expected zero-crossing rates given by κ/π√(-ρ"(0)) with either κ≫1 or κ≪1. Consequently, such processes can have an arbitrarily large or small zero-crossing rate as compared to a Gaussian process with the same autocorrelation function