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This chapter considers various models that focus largely on serially dependent variables and the respective methodologies developed with a COM–Poisson underpinning. This chapter first introduces the reader to the various stochastic processes that have been established, including a homogeneous COM–Poisson process, a copula-based COM–Poisson Markov model, and a COM–Poisson hidden Markov model. Meanwhile, there are two approaches for conducting time series analysis on time-dependent count data. One approach assumes that the time dependence occurs with respect to the intensity vector. Under this framework, the usual time series models that assume a continuous variable can be applied. Alternatively, the time series model can be applied directly to the outcomes themselves. Maintaining the discrete nature of the observations, however, requires a different approach referred to as a thinning-based method. Different thinning-based operators can be considered for such models. The chapter then broadens the discussion of dependence to consider COM–Poisson-based spatio-temporal models, thus allowing both for serial and spatial dependence among variables.
Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation for generators is equivalent to Fristedt's Lévy measure factorisation and to process decomposition.
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