Markoff-Ketten bei sich füllenden Löchern im Zustandsraum
Annales de l'Institut Fourier, Volume 21 (1971) no. 1, pp. 253-270.

Given a substochastic kernel $P$ from a measurable space $\left(E,\beta \right)$ into itself one considers for a pair $\left(\mu ,\nu \right)$ of finite measures on $\beta$ the following sequences:

 ${\mu }_{0}=\left(\mu -\nu {\right)}^{+},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\nu }_{0}=\left(\mu -\nu {\right)}^{-}\phantom{\rule{3.33333pt}{0ex}};$
 ${\mu }_{n+1}=\left({\mu }_{n}^{p}-{\nu }_{n}{\right)}^{+},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\nu }_{n+1}=\left({\mu }_{n}^{p}-{\nu }_{n}{\right)}^{-},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n=0,1,2,...$

This paper deals with conditions for $↓{lim}_{n}{\nu }_{n}=0$, or ${lim}_{n}\parallel {\mu }_{n}\parallel =0$ to hold. As an application a characterization of those measures $\nu$ is given which may occur in a $P$-Markov chain $\left({X}_{n}{\right)}_{n\in \mathbf{N}}$ with state space $E$, having $\mu$ as its initial law, as distribution of ${X}_{\tau }$ where $\tau$ is a suitable stopping time.

Soit $P$ un noyau substochastique d’un espace mesurable $\left(E,\beta \right)$ dans lui-même ; pour chaque paire $\left(\mu ,\nu \right)$ de mesures finies sur $\beta$, on considère les suites suivantes :

 ${\mu }_{0}=\left(\mu -\nu {\right)}^{+},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\nu }_{0}=\left(\mu -\nu {\right)}^{-}\phantom{\rule{3.33333pt}{0ex}};$
 ${\mu }_{n+1}=\left({\mu }_{n}^{p}-{\nu }_{n}{\right)}^{+},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\nu }_{n+1}=\left({\mu }_{n}^{p}-{\nu }_{n}{\right)}^{-},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n=0,1,2,...$

Cet article traite des conditions pour que l’on ait $↓{lim}_{n}{\nu }_{n}=0$, ou ${lim}_{n}\parallel {\mu }_{n}\parallel =0$.

Comme application, on donne une caractérisation des mesures $\nu$, qui dans une $P$-chaîne de Markov $\left({X}_{n}{\right)}_{n\in \mathbf{N}}$ dont $E$ est l’espace des états, de loi initiale $\mu$, peuvent apparaître comme distribution de ${X}_{\tau }$$\tau$ est un temps d’arrêt adéquat.

@article{AIF_1971__21_1_253_0,
author = {Rost, Hermann},
title = {Markoff-Ketten bei sich f\"ullenden {L\"ochern} im {Zustandsraum}},
journal = {Annales de l'Institut Fourier},
pages = {253--270},
publisher = {Institut Fourier},
volume = {21},
number = {1},
year = {1971},
doi = {10.5802/aif.366},
mrnumber = {45 #8803},
zbl = {0197.44206},
language = {de},
url = {http://www.numdam.org/articles/10.5802/aif.366/}
}
TY  - JOUR
AU  - Rost, Hermann
TI  - Markoff-Ketten bei sich füllenden Löchern im Zustandsraum
JO  - Annales de l'Institut Fourier
PY  - 1971
SP  - 253
EP  - 270
VL  - 21
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.366/
DO  - 10.5802/aif.366
LA  - de
ID  - AIF_1971__21_1_253_0
ER  - 
%0 Journal Article
%A Rost, Hermann
%T Markoff-Ketten bei sich füllenden Löchern im Zustandsraum
%J Annales de l'Institut Fourier
%D 1971
%P 253-270
%V 21
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.366/
%R 10.5802/aif.366
%G de
%F AIF_1971__21_1_253_0
Rost, Hermann. Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. Annales de l'Institut Fourier, Volume 21 (1971) no. 1, pp. 253-270. doi : 10.5802/aif.366. http://www.numdam.org/articles/10.5802/aif.366/

[1] M.A. Akcoglu, R.W. Sharpe, Ergodic theory and boundaries, Trans. Amer. Math. Soc. 132 (1968), 447-460. | MR | Zbl

[2] R.V. Chacon, D.S. Ornstein, A general ergodic theorem, III. Journal Math. 4 (1960), 153-160. | MR | Zbl

[3] E.B. Dynkin, A.A. Juschkewitsch, Sätze und Aufgaben über Markoffsche Prozesse, Berlin-Heidelberg-New York : Springer 1969. | Zbl

[4] P.A. Meyer, Théorie ergodique et potentiels, Ann. Inst. Fourier 15 (1965), 89-102. | Numdam | MR | Zbl

[5] P.A. Meyer, Probabilités et potentiel, Paris : Hermann 1966. | MR | Zbl

[6] H. Rost, Darstellung einer Ordnung von Massen durch Stoppzeiten, Z. Wahrscheinlichkeitstheorie verw. Geb. 15 (1970), 19-28. | MR | Zbl

Cited by Sources: