Rough invariance principle for delayed regenerative processes

Tal Orenshtein

doi:10.1214/21-ecp406

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 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.

Highlights

Donsker’s invariance principle states that a diffusively rescaled centered random walk on Rd with jumps of finite variance converges in distribution to a Brownian motion in the Skorohod topology
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
The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition

Summary

Introduction

Donsker’s invariance principle states that a diffusively rescaled centered random walk on Rd with jumps of finite variance converges in distribution to a Brownian motion in the Skorohod topology. The first level is naturally, the Brownian motion defined by the covariance matrix achieved in the classical case, whereas the second level (see Section 1.1 for more details) does not coincide with the iterated integral of the Brownian motion, but should be corrected by a deterministic process The latter is called ‘area anomaly’ and is linear in time and identified in terms of the stochastic area on a regeneration time interval, see for example (5) below. In order to optimize the moment condition on a regeneration interval for processes with regenerative increments so that the rough path version of Donsker’s Theorem is used under not more than the second moment, we apply the Key Renewal Theorem This theorem roughly says that the mass function of the process in a regeneration interval around a fixed time ( called ‘age’ in the renewal theory jargon) is approaching a density which is proportional to the uniform measure of an independent copy of the interval, namely, its size-biased version. We included three short appendices which might be useful in other context

Preliminaries on rough paths

Main result

Examples

A Key renewal theorem

B Slutsky’s Theorem in the Skorohod topology

C Rough Donsker Theorem