Method to get the mean recurrence times \(\mu\).
meanRecurrenceTimes(x, klim = 10000)An object of S3 class smmfit or smm.
Optional. The time horizon used to approximate the series in the computation of the mean sojourn times vector \(m\) (cf. meanSojournTimes function).
A vector giving the mean recurrence time \((\mu_{i})_{i \in [1,\dots,s]}\).
Consider a system (or a component) \(S_{ystem}\) whose possible states during its evolution in time are \(E = \{1,\dots,s\}\).
We are interested in investigating the mean recurrence times of a discrete-time semi-Markov system \(S_{ystem}\). Consequently, we suppose that the evolution in time of the system is governed by an E-state space semi-Markov chain \((Z_k)_{k \in N}\). The state of the system is given at each instant \(k \in N\) by \(Z_k\): the event \(\{Z_k = i\}\).
Let \(T = (T_{n})_{n \in N}\) denote the successive time points when state changes in \((Z_{n})_{n \in N}\) occur and let also \(J = (J_{n})_{n \in N}\) denote the successively visited states at these time points.
The mean recurrence of an arbitrary state \(j \in E\) is given by:
$$\mu_{jj} = \frac{\sum_{i \in E} \nu(i) m_{i}}{\nu(j)}$$
where \((\nu(1),\dots,\nu(s))\) is the stationary distribution of the embedded Markov chain \((J_{n})_{n \in N}\) and \(m_{i}\) is the mean sojourn time in state \(i \in E\) (see meanSojournTimes function for the computation).