Tied-down occupation times of infinite ergodic transformations

Abstract

We prove distributional limit theorems (conditional and integrated) for the occupation times of certain weakly mixing, pointwise dual ergodic transformations at “tied-down” times immediately after “excursions”. The limiting random variables include the local times of q-stable Lévy-bridges (1 < q ≤ 2) and the transformations involved exhibit “tied-down renewal mixing” properties which refine rational weak mixing. Periodic local limit theorems for Gibbs—Markov and AFU maps are also established.