Quasi-Ergodic Limits for Moments of Jumps Under Absorbing Stable Processes

We study a quasi-ergodic theorem for moments of the mean-ratio of a discontinuous additive functional caused by the pure jump effects of an absorbing symmetric stable process on an open set \(D \subset \mathbb {R}^d\) $D\subset {\mathbb{R}}^d$ . Our result depends only on a simple geometric condition on D and does not require that the total mass of D is finite. As a result, we prove that the result holds if D is a subset of a horn-shaped region with the reference function satisfying some condition.