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Transactions of the American Mathematical Society

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$ \lambda $-continuous Markov chains. II


Author: Shu-teh C. Moy
Journal: Trans. Amer. Math. Soc. 120 (1965), 83-107
MSC: Primary 60.65
DOI: https://doi.org/10.1090/S0002-9947-1965-0183020-8
MathSciNet review: 0183020
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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 $.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9947-1965-0183020-8
Article copyright: © Copyright 1965 American Mathematical Society

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