Pedestrian's approach to large deviations in semi-Markov processes with an application to entropy production

Semi-Markov processes play an important role in the effective description of partially accessible systems in stochastic thermodynamics. They occur, for instance, in coarse-graining procedures such as state lumping and when analyzing waiting times between few visible Markovian events. The finite-time measurement of any coarse-grained observable in a stochastic system depends on the specific realization of the underlying trajectory. Moreover, the fluctuations of such observables are encoded in their rate function that follows from the rate function of the empirical measure and the empirical flow in the respective process. Derivations of the rate function of empirical measure and empirical flow in semi-Markov processes in which the sojourn time in a state is independent of the next state, a property called direction-time independence (DTI), exist in the mathematical literature, but have not received much attention in the stochastic thermodynamics community. We present an accessible derivation of the rate function of the tuple frequency, i.e., the frequency of two consecutively observed states, in discrete-time Markov chains, and extend this to the rate function of the empirical semi-Markov kernel in semi-Markov processes without DTI. From this, we derive an upper bound on the rate function of the empirical entropy production rate, which leads to a lower bound on the variance of the mean entropy production rate measured along a finite-time trajectory. We illustrate these analytical bounds with simulated data.