  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 first
main result is a necessary and sufficient condition 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. Our condition is considerably weaker than that needed
for ergodic behaviour; a similar notion which demands that marginal upper and
lower expectations of functions at a single time instant converge to so-called
limit-or steady state-upper and lower expectations. For this reason, we refer
to our notion as `weak ergodicity'. Our second main result shows that, as far
as this weakly ergodic behaviour is concerned, one should not worry about which
type of independence assumption to adopt-epistemic irrelevance, complete
independence or repetition independence. The characterisation of weak
ergodicity as well as the limit values of upper and lower expected time
averages do not depend on such a choice. Notably, this type of robustness is
not exhibited by the notion of ergodicity and the related inferences of limit
upper and lower expectations. Finally, though limit upper and lower
expectations are often used to provide approximate information about the limit
behaviour of time averages, 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.
