$\lambda$-continuous Markov chains. II

Abstract:

Continuing the investigation in [8] we study a $\lambda$-continuous Markov operator $P$. It is shown that, if $P$ is conservative and ergodic, $P$ is indeed “periodic” as is the case when the state space is discrete; there is a positive integer $\delta$, called the period of $P$, such that the state space may be decomposed into $\delta$ cyclically moving sets ${C_0}, \cdots ,{C_{\delta - 1}}$ and, for every positive integer $n,{P^{n\delta }}$ acting on each ${C_i}$ alone is ergodic. It is also shown that $P$ maps ${L_q}(\mu )$ into ${L_q}(\mu )$ where $\mu$ is the nontrivial invariant measure of $P$ and $1 \leqq q \leqq \infty$. If $\mu$ is finite and normalized then it is shown that (1) if $f \in {L_\infty }(\lambda )$, then $\{ {P^{n\delta + k}}f\}$ converges a.e. $(\lambda )$ to ${g_k} = \sum \nolimits _{i = 0}^{\delta - 1} {{c_{i + k}}} {1_{{C_i}}}$ where ${c_j} = \delta {\smallint _{{C_j}}}fd\mu$ if $0 \leqq j \leqq \delta - 1$ and ${c_j} = {c_i}$ if $j = m\delta + i,0 \leqq i \leqq \delta - 1$, (2) $\{ {P^{n\delta + k}}f\}$ converges in ${L_q}(\mu )$ to ${g_k}$ if $f \in {L_q}(\mu )$, and(3) $\lim {\inf _{n \to \infty }}{P^{n\delta + k}}f = {g_k}$ a.e. $(\lambda )$ if $f \in {L_1}(\mu )$ and $f \geqq 0$. If $\mu$ is infinite, then it is shown that (1) if $f \geqq 0,f \in {L_q}(\mu )$ for some $1 \leqq q < \infty$, then $\lim {\inf _{n \to \infty }}{P^n}f = 0$ a.e. $(\lambda )$, (2) there exists a sequence $\{ {E_k}\}$ of sets such that $X = \cup _{k = 1}^\infty {E_k}$ and ${\lim _{n \to \infty }}{P^{n\delta + i}}{1_{{E_k}}} = 0$ a.e. $(\lambda )$ for $i = 0,1, \cdots ,\delta - 1$ and $k = 1,2, \cdots$.