  We study the limit behaviour of upper and lower bounds on expected time
averages in imprecise Markov chains; a generalised type of Markov chain where
the local dynamics, traditionally characterised by transition probabilities,
are now represented by sets of 'plausible' transition probabilities. Our main
result is a set of necessary and sufficient conditions under which these upper
and lower bounds, called upper and lower expected time averages, will converge
as time progresses towards infinity to limit values that do not depend on the
process' initial state. Remarkably, our conditions are considerably weaker than
those needed to establish similar results for so-called limit -- or steady
state -- upper and lower expectations, which are often used to provide
approximate information about the limit behaviour of time averages as well. We
show that such an approximation is sub-optimal and that it can be significantly
improved by directly using upper and lower expected time averages.
