Convergence Parameter Associated with a Markov Chain and a Family of Functions

The proposed definition of convergence parameter $R(W)$ corresponding to a Markov chain $X$ with a measurable state space $(E,\mathscr B)$ and any nonempty set $W$ of bounded below measurable functions $f\colon E\to\mathbb R$ is wider than the well-known definition of convergence parameter $R$ in the sense of Tweedie or Nummelin. Very often, $R(W)<\infty$, and there exists a set playing the role of the absorbing set in Nummelin's definition of $R$. Special attention is paid to the case in which $E$ is locally compact, $X$ is a Feller chain on $E$, and $W$ coincides with the family $\mathscr C_0^+$ of all compactly supported continuous functions $f\ge 0$ ($f\not\equiv 0$). In particular, certain conditions for $R(\mathscr C_0^+)^{-1}$ to coincide with the norm of an appropriate modification of the chain transition operator are found.