## The Annals of Probability

### Backward Limits

Hermann Thorisson

#### 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.

#### Article information

Source
Ann. Probab., Volume 16, Number 2 (1988), 914-924.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991796

Digital Object Identifier
doi:10.1214/aop/1176991796

Mathematical Reviews number (MathSciNet)
MR929087

Zentralblatt MATH identifier
0643.60033

JSTOR