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 ß. We then focus on ß?(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 ß.