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.

我们研究了不精确马尔可夫链中期望时间平均值的上限和下限的极限行为；马尔可夫链的一种广义类型，传统上以转移概率为特征的局部动力学现在由“似然”转移概率集表示。我们的第一个主要结果是一个必要的充分条件，在这种条件下，这些上限和下限（称为上限和下限预期时间平均值）将随着时间逐渐趋近无穷而收敛，以限制不依赖于过程初始状态的值。我们的条件比遍历行为所需的条件要弱得多。类似的概念要求在单个时刻对功能的边际上，下期望值收敛到所谓的极限（或稳态）上，下期望值。为此原因，我们将我们的概念称为“弱遍历”。我们的第二个主要结果表明，就这种弱遍历行为而言，人们不应该担心采用哪种类型的独立性假设—流行病无关性，完全独立性或重复独立性。弱遍历性的表征以及预期时间平均值的上限和下限均不取决于这种选择。值得注意的是，遍历性的概念以及有关上限和下限期望的相关推论都没有表现出这种鲁棒性。最后，尽管通常使用极限上限和下限期望来提供有关时间平均值的极限行为的近似信息，