On the Markov-switching bilinear processes: stationarity, higher-order moments and β-mixing
This article investigates some probabilistic properties and statistical applications of general Markov-switching bilinear processes
that offer remarkably rich dynamics and complex behaviour to model non-Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class
of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and -mixing
are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (respectively, higher)-order
structure is similar to some linear processes, and hence admit representation. We establish also sufficient conditions for the
model to be mixing
and geometrically ergodic. We then use these results to give sufficient conditions for mixing of a family of processes. A number of illustrative examples are given to clarify the theory and the variety of applications.
Keywords: Markov-switching bilinear processes; Primary 62M10; Secondary 62M05; geometric ergodicity; mixing; representation; stationarity
Document Type: Research Article
Affiliations: Département de Mathématiques, Université Constantine(1), Constantine, Algeria
Publication date: 02 November 2015
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