RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 380–383 (Mi tvp2605)

Short Communications

A generalization of an ergodic theorem of Hopf

A. A. Tempel'man

Institute of Physics and Mathematics, Academy of Sciences, Lithuanian SSR

Abstract: Let $X$ be a separable locally compact semigroup; let($\Omega$, $\mathfrak G$, $m$) be a space with a $\sigma$-finite measure $m$ and let $T_x$, $x\in X$, be a dynamic system in $\Omega$ with “time” from $X$. Let, further, $p$ and $q$ be probability Borel measures on $X$ and $\lambda_n=\sum_{k=0}^np*q^{*k}$. If $f$, $g\in L_1(m)$ and $g>0$ then the limit
$$\lim_{n\to\infty}\int_Xf(T_x\omega)\lambda_n(dx)/\int_Xg(T_x\omega)\lambda_n(dx)=h_{f,g}(\omega)$$
is shown to exist almost everywhere on $\Omega$.
$(p,q)$-conservative dynamic systems are defined as systems inducing recurrent random walks in $\Omega$ in correspondence with the measures $p$ and $q$. For such dynamic systems the equality $h_{f,g}=\mathbf E(f\mid\mathfrak F)$ is proved where $\mathbf E(f\mid\mathfrak F)$ is the conditional expectation of the function $f(\omega)$ given the $\sigma$-algebra $\mathfrak F$ of measurable invariant sets.

Full text: PDF file (272 kB)

English version:
Theory of Probability and its Applications, 1973, 17:2, 363–365

Bibliographic databases:

Citation: A. A. Tempel'man, “A generalization of an ergodic theorem of Hopf”, Teor. Veroyatnost. i Primenen., 17:2 (1972), 380–383; Theory Probab. Appl., 17:2 (1973), 363–365

Citation in format AMSBIB
\Bibitem{Tem72} \by A.~A.~Tempel'man \paper A~generalization of an ergodic theorem of Hopf \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 2 \pages 380--383 \mathnet{http://mi.mathnet.ru/tvp2605} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=301169} \zmath{https://zbmath.org/?q=an:0265.28008} \transl \jour Theory Probab. Appl. \yr 1973 \vol 17 \issue 2 \pages 363--365 \crossref{https://doi.org/10.1137/1117043}