## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 5 (1981), 781-799.

### Splitting at Backward Times in Regenerative Sets

#### Abstract

By a backward time is meant a random time which only depends on the future, in the same sense as a stopping time only depends on the past. We show that backward times taking values in a regenerative set $M$ split $M$ into conditionally independent subsets. The conditional distributions of the past may further be identified with the Palm distributions $P_t$ with respect to the local time random measure $\xi$ of $M$ both a.e. $E\xi$ and wherever $\{P_t\}$ has a continuous version. Continuity of $\{P_t\}$ occurs essentially where $E\xi$ has a continuous density, and the latter continuity set may be described rather precisely in terms of the growth rate and regularity properties of the Levy measure of $M$.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 5 (1981), 781-799.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994308

**Digital Object Identifier**

doi:10.1214/aop/1176994308

**Mathematical Reviews number (MathSciNet)**

MR628873

**Zentralblatt MATH identifier**

0526.60061

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces

Secondary: 60J55: Local time and additive functionals 60K05: Renewal theory 60G57: Random measures

**Keywords**

Conditional independence regenerative set local time Palm distribution renewal density

#### Citation

Kallenberg, Olav. Splitting at Backward Times in Regenerative Sets. Ann. Probab. 9 (1981), no. 5, 781--799. doi:10.1214/aop/1176994308. https://projecteuclid.org/euclid.aop/1176994308