  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 $\beta$.
