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 $M$.
Publié le : 1981-10-14
Classification:
Conditional independence,
regenerative set,
local time,
Palm distribution,
renewal density,
60J25,
60J55,
60K05,
60G57
@article{1176994308,
author = {Kallenberg, Olav},
title = {Splitting at Backward Times in Regenerative Sets},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 781-799},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994308}
}
Kallenberg, Olav. Splitting at Backward Times in Regenerative Sets. Ann. Probab., Tome 9 (1981) no. 6, pp. 781-799. http://gdmltest.u-ga.fr/item/1176994308/