The principal semi-algebra in a Banach algebra
Edward J.
Barbeau
1-16
Representations of graded Lie algebras
Leonard E.
Ross
17-23
Generalized Fekete means
Paul
Schaefer
24-36
Homotopy properties of the space of homeomorphisms on $P\sp{2}$ and the Klein bottle
Mary-Elizabeth
Hamstrom
37-45
Normal curves arising from light open mappings of the annulus
Morris L.
Marx
46-56
On the representations of an abstract lattice as the family of closed sets of a topological space
David
Drake;
W. J.
Thron
57-71
Multivalent functions star-like in one direction
Teruo
Takatsuka
72-82
$\lambda $-continuous Markov chains. II
Shu-teh C.
Moy
83-107
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 $k = 1,2, \cdots$.
Convergence rates in the law of large numbers
Leonard E.
Baum;
Melvin
Katz
108-123
Generalized Taylor series and orders and types of entire functions of several complex variables
Fred
Gross
124-144
An eigenvalue problem for nonlinear elliptic partial differential equations
Melvyn S.
Berger
145-184