The Marginal Distribution Function of Threshold-type Processes with Central Symmetric Innovations

This paper addresses the problem of finding exact and explicit (closed-form) expressions for the stationary marginal distribution of threshold-type time series processes, their associated moments, autocovariance and autocorrelation coefficients. The innovation process of the models under consideration follow three central symmetric distribution functions: Gaussian, Laplace, and Cauchy. Theoretical results for both two- and three regime threshold-type models are derived. Various examples give rise to a deeper understanding of certain features of the stationary process structure. Exact results for the stationary density, central moments, and autocorrelations of threshold-type processes are compared with approximate density and moment results obtained through an existing numerical method.