Stochastic Renewal Equation for the Waiting Time Statistics for the First Occurrence of a Specific Sequence of States Successively Visited by an Alternating Renewal Process

Both Markovian and arbitrary residence time distributions are considered. The comparison of analytical predictions with the case of a time-decorrelated process shows that correlations can both decrease and increase the corresponding expected waiting time. Besides, the comparison of exponential, subexponential, and heavy-tailed models characterized by equal probabilities to observe the event of interest demonstrates that a faster decrease in the residence time probability density implies a shorter expected waiting time. Interestingly, irrespective of the details of a particular model for both discrete- and continuous-time jump processes considered here, the random waiting time becomes exponentially distributed in the long-time limit, thus, showing remarkable universality.