A point process (or counting process) is a type of random process for which any generic realization consists of a set of isolated points either in time or geographical space, or in even more general spaces. Generally, in order to describe the event (point) occurrences over time, a point process relies on the sequential arrival points over (0,∞). On the other hand, a counting process describes the random occurrences of events in terms of the counts of events observed over time intervals. In the following, for convenience, we will use the terms ‘point process’ and ‘counting process’ interchangeably. Until now, various univariate point processes have been proposed and the properties of these have been intensively studied. The univariate point processes which are most frequently applied in practice include the Markov process, homogeneous and nonhomogeneous Poisson processes, and the renewal process. In particular, homogeneous and nonhomogeneous Poisson processes are the most commonly used point processes in practice due to their nice properties, which generally yield explicit results in most applications even when the stochastic model under study is complex; see, for example, Cha and Finkelstein (2009), Cha and Lee (2011), and Liang andWang (2012).
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