## The Annals of Mathematical Statistics

### An Operator Theorem on $L_1$ Convergence to Zero with Applications to Markov Kernels

#### Abstract

A recent theorem of Orey  (see also , , , ) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

#### Article information

Source
Ann. Math. Statist., Volume 41, Number 5 (1970), 1631-1639.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177696806

Digital Object Identifier
doi:10.1214/aoms/1177696806

Mathematical Reviews number (MathSciNet)
MR272057

Zentralblatt MATH identifier
0284.60068

JSTOR
Ornstein, Donald; Sucheston, Louis. An Operator Theorem on $L_1$ Convergence to Zero with Applications to Markov Kernels. Ann. Math. Statist. 41 (1970), no. 5, 1631--1639. doi:10.1214/aoms/1177696806. https://projecteuclid.org/euclid.aoms/1177696806