Backward Limits

Abstract

We consider a time-inhomogeneous regenerative process starting from regeneration at time $s$ and prove, under regularity conditions on the regeneration times, that the distribution of the process in a fixed time interval $\lbrack t, \infty)$ stabilizes as the starting time $s$ tends backward to $-\infty$ (the convergence considered here is in the sense of total variation). This implies the existence of a two-sided time-inhomogeneous process "starting from regeneration at $-\infty$." We also show that if a time-inhomogeneous regenerative process admits a limit law in the traditional forward sense, then it is asymptotically time-homogeneous; thus the backward approach widely extends the class of processes admitting a limit law.