Theoretical analysis reveals criteria for stationary regime uniqueness in stochastically monotone Markov chains, suggesting broader applicability beyond uniform swap bounds.
For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (2007) where a certain form of mixing, or swap condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities.
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Foss et al. (2024) studied this question.