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 path version of Donsker’s Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.
This work was supported by the German Research Foundation (DFG) via Research Unit FOR2402.
The author is grateful to Tommaso Cornelis Rosati and Willem van Zuijlen for numerous valuable discussions and to the anonymous referee for useful remarks.
"Rough invariance principle for delayed regenerative processes." Electron. Commun. Probab. 26 1 - 13, 2021. https://doi.org/10.1214/21-ECP406