Properties of stationary cyclical processes

Abstract

The paper investigates the theoretical properties of zero-mean stationary time series with cyclical components, admitting the representation $y_t=\alpha_t \cos \lambda t + \beta_t \sin \lambda t$, with $\lambda \in (0,\pi]$ and $[\alpha_t\,\, \beta_t]$ following some bivariate process. We diagnose that in the extant literature on cyclic time series, a prevalent assumption of Gaussianity for $[\alpha_t\,\, \beta_t]$ imposes inadvertently a severe restriction on the amplitude of the process. Moreover, it is shown that other common distributions may suffer from either similar defects or fail to guarantee the stationarity of $y_t$. To address both of the issues, we propose to introduce a direct stochastic modulation of the amplitude and phase shift in an almost periodic function. We prove that this novel approach may lead, in general, to a stationary (up to any order) time series, and specifically, to a zero-mean stationary time series featuring cyclicity, with a pseudo-cyclical autocovariance function that may even decay at a very slow rate. The proposed process fills an important gap in this type of models and allows for flexible modeling of amplitude and phase shift.