The Annals of Probability

Splitting at Backward Times in Regenerative Sets

Olav Kallenberg

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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

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

First available in Project Euclid: 19 April 2007

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Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J55: Local time and additive functionals 60K05: Renewal theory 60G57: Random measures

Conditional independence regenerative set local time Palm distribution renewal density


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

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