Continuity of imprecise Markov chains with respect to the pointwise convergence of monotone sequences of gambles

The aim of these notes is to study the conditions under which the natural extension of an imprecise Markov chain is continuous with respect to the pointwise convergence of monotone (either non-decreasing or non-increasing) sequences of gambles that are $n$-measurable, with $n\in\mathbb{N}_{0}$. The framework in which we do this is that of the theory of random processes that is being developed in [de Cooman, 2014]. We find that for non-decreasing sequences, continuity is always guaranteed if the state space of the markov chain is finite. A similar result is obtained for non-increasing sequences, under the additional condition that the joint model is constructed using the Ville-Vovk-Shafer natural extension rather than the Williams natural extension. The extent to which these results apply to general random processes is discussed as well.