Open Access
January, 1987 Homogeneity and the Strong Markov Property
Olav Kallenberg
Ann. Probab. 15(1): 213-240 (January, 1987). DOI: 10.1214/aop/1176992265


The strong Markov property of a process $X$ at a stopping time $\tau$ may be split into a conditional independence part (CI) and a homogeneity part (H). However, it turns out that (H) often implies at least some version of (CI). In the present paper, we shall assume that (H) holds on the set $\{X_\tau \in B\}$, for all stopping times $\tau$ such that $X_\tau \in F$ a.s., where $F$ is a closed recurrent subset of the state space $S$, while $B \subset F$. If $F = S$, then (CI) will hold on $\{X_\tau \in B\}$ for every stopping time $\tau$, so in this case $X$ is regenerative in $B$. In the general case, the same statement is conditionally true in a suitable sense, given some shift invariant $\sigma$-field.


Download Citation

Olav Kallenberg. "Homogeneity and the Strong Markov Property." Ann. Probab. 15 (1) 213 - 240, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0614.60063
MathSciNet: MR877599
Digital Object Identifier: 10.1214/aop/1176992265

Primary: 60J25
Secondary: 60G40

Keywords: Conditional independence , exchangeable sequences , invariant $\sigma$-fields , recurrence , regeneration , stopping times

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
Back to Top