Bounds for mixing times for finite semi-Markov processes with heavy-tail jump distribution

Consider a Markov chain with finite state space and suppose you wish to change time replacing the integer step index $n$ with a random counting process $N(t)$. What happens to the mixing time of the Markov chain? We present a partial reply in a particular case of interest in which $N(t)$ is a counting renewal process with power-law distributed inter-arrival times of index $\beta$. We then focus on $\beta \in (0,1)$ , leading to infinite expectation for inter-arrival times and further study the situation in which inter-arrival times follow the Mittag-Leffler distribution of order $\beta$.