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 $M$.