Rough invariance principle for delayed regenerative processes

TL;DR

This paper establishes an invariance principle for delayed regenerative processes in the rough path topology, revealing an area anomaly correction term and applying to random walks and Markov chains under optimal moment conditions.

Contribution

It introduces a rough path invariance principle for delayed regenerative processes with an explicit area anomaly, under optimal moment conditions, extending classical results.

Findings

Invariance principle for delayed regenerative processes in rough path topology.

Identification of an area anomaly as the average stochastic area.

Applicability to random walks in random environments and Markov chains.

Abstract

We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.