A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

Continuous-time Markov chain models have been widely considered for the gating behaviour of a single ion channel. In such models the state space is usually partitioned into two classes, designated ‘open' and ‘closed', and there is ‘aggregation' in that it is possible to observe only which class the process is in at any given time. Hawkes et al. (1990) have derived an expression for the density function of the exact distribution of an observed open time in such an aggregated Markov model, where brief sojourns in either the open or the closed class are unobservable. This paper extends their result to single ion channel models based on aggregated semi-Markov processes, giving a more direct derivation which is probabilistic and exhibits clearly the combinatorial content.

We consider a two-state Markov process in which the resolution of the recording apparatus is such that small sojourns, of duration less than some constant deadtime τ, cannot be observed: the so-called time interval omission problem. We express the probability density of apparent occupancy times in terms of an exponential and infinitely many damped oscillations. Using a finite number of these gives an extremely accurate approximation to the true density for all except small values of the time t.