Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ associated with positive real numbers $\epsilon(\mathfrak{S})$ and a minorizing probability measure $\nu$ such that $$ \mathbb{P}(X_{t+1}\in A|X_t) \ge \epsilon(\mathfrak{S}) \nu(A) $$ whenever $X_t\in \mathfrak{S}$.

If I call renewal rate associated with a small set $\mathfrak{S}$ the quantity $$ \varrho(\mathfrak{S}) = \epsilon(\mathfrak{S})\nu(\mathfrak{S})\,, $$ is there an intrinsic definition or a characterisation of the maximal renewal rate $$ \max_\mathfrak{S} \varrho(\mathfrak{S}) $$ and is it related to the convergence properties of the Markov chain $(X_t)$?



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