Markov chains efficiently describe the uncertain time evolution of various systems and are a very popular tool in a wide range of applied domains, including queueing theory, artificial intelligence, bioinformatics, operations research and reliability engineering. The simplifying (Markov) assumption of these models is not always realistic, though, nor is it realistic to specify precise accurate values for their parameters. This may adversely impact the validity of the conclusions drawn from them. That is where imprecise Markov chains come in, which allow for partially specified parameters. They characterise a wide range of general (not necessarily Markovian) stochastic processes whose dynamics are compatible with this partial specification, and they can be used to obtain (tight) bounds on the predicted future behaviour for all of these compatible stochastic processes. How to determine such bounds has been studied mainly for the expected behaviour of processes, but much less so for their actual behaviour. In particular, one is often interested in the average behaviour of a process along the actual path it takes, for those times when a specific event happens. For example, the frequency of having no available bed on the days where a new patient needs one. This project aims to develop the mathematical foundations for dealing with such inferences, using so-called ergodic theorems, and to develop algorithms for calculating the (tight) bounds that these theorems provide.