First of all: I'm talking about Markov chains over finite spaces. Second: my grasp of formal notation is rather weak, and in general I'm pretty ignorant, so please bear with me. That said...
All the (admittedly not too many) definitions of mixing time I've seen, including the total variation distance mixing time and Cesaro mixing time, consider measures of deviation from the stationary probability that are essentially additive. So, if an event has a really small probability in the stationary distribution, convergence is still considered to occur even with a disproportionally large relative error. This is a problem if I am considering e.g. Monte Carlo simulations of catastrophic events: there's a lot of "difference" from my point of view between an event occurring with probability $10^{-60}$ and one occurring with probability $10^{-6}$; there is far less "difference" between an event occurring with probability 0.1 and one occurring with probability 0.1001.
Is there any well known definition of mixing time that uses multiplicative error, such as the time after which the probability of each state has converged within a factor $2$ of the stationary probability?