Two Theorems on Convergence Parameter of an Irreducible Markov Chain
M. G. Shur · Theory of Probability and Its Applications · 2014
A homogeneous irreducible Markov chain $X$ with a measurable state space $(E,{\cal B})$ and a transient operator $P$ acting in a probability of bounded from below measurable functions is considered. The $\sigma$-algebra ${\cal B}$ is assumed to be countably generated. It is proved that if the chain is aperiodic and function $f$ and measure $ u$ are small, then $[ u(P^nf)]^{1/n}\rightarrow R$ as $n\rightarrow\infty$, where $R$ is the convergence parameter. For periodic Markov chains this statement can be modified in the following way. If a chain $X$ is symmetric with respect to some $\sigma$-finite measure $\pi$, then $R=\|\widetilde{P}\|^{-1}$, where $\widetilde{P}$ is a bounded self-adjoint operator generated by $P$ and acting in the space $L_2 (\pi)$. Results of this paper extend the results of M. G. Shur [Math. Notes, 75 (2004), pp. 864--876; Math. Notes, 87 (2010), pp. 271--280].