We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.