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The batch Markov‐modulated Poisson process (BMMPP) is a subclass of the versatile batch Markovian arrival process (BMAP), which has been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures, or risk events). Both theoretical and applied aspects are examined in this paper. On one hand, the identifiability of the stationary BMMPPm(K ) is proven, where K is the maximum batch size and m is the number of states of the underlying Markov chain. This is a powerful result for inferential issues. On the other hand, some novelties related to the correlation and autocorrelation structures are provided.

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This dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the (Batch) Markov Modulated Poisson process. Point processes arise in a wide range of situations in physics, biology, engineering, or economics. In general, the occurrence of events is defined depending on the context, but in many areas are defined by the occurrence of an event at a specific time. Sometimes, in order to simplify the models and obtain closed form expressions for the quantities of interest, the exponentiality and/or independence of the inter-event times is assumed. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice, and therefore, there is a need of more realistic models to fit the data.