Functional Scaling Limits of Interpolated Correlated Random Walks in H\"older Topology

Abstract

We prove functional scaling limits for interpolated random walks whose increments are functions of a stationary Gaussian sequence. In this setting, the classical Dobrushin-Major-Taqqu theorem describes the scaling limit when the covariance has a regularly varying, non-summable tail, while the Breuer-Major theorem describes the limit in the summable regime. We strengthen these convergence results to functional convergence in H\"older topology and, in the summable regime, in rough H\"older topology. These stronger topologies are useful because many operations on paths, such as Young integration and solution maps of differential equations, are continuous in (rough) H\"older topology, but not in Skorokhod topology.