Statistical Inference for Homogenization Limits Driven by Wiener or Hermite Processes

TL;DR

This paper develops methods to estimate diffusivity and Hurst parameters in slow/fast systems with limits driven by Wiener or Hermite processes, ensuring estimator consistency under subsampling.

Contribution

It introduces a subsampling framework that preserves estimator consistency for homogenized limits driven by Wiener or Hermite processes, using Wiener chaos expansions.

Findings

Estimators remain consistent under appropriate subsampling.

A non-central limit theorem is preserved with stricter subsampling.

Consistency of a self-similarity estimator is established without knowing diffusivity.

Abstract

We study the effective estimation of the diffusivity and Hurst parameter for the homogenized limit of a class of slow/fast systems. Depending on the system parameters, this limit solves a stochastic differential equation driven by either a Wiener process or a Hermite process. In the class of models we consider, the fast variable is a fractional Ornstein--Uhlenbeck process. We show that estimators constructed from the homogenized limit remain consistent when applied to appropriately subsampled data generated by the original slow/fast system. A key tool in our analysis is the consistency of renormalized quadratic variations for a family of additive functionals of the fast process. Using Wiener chaos expansions, we obtain an \(L^2\)-orthogonal decomposition of these renormalized quadratic variations. This allows us to show that, under appropriate subsampling conditions, the consistency properties of the estimators are preserved even when the data is generated by the slow/fast system rather than the homogenized limit. We also show that, under stricter subsampling conditions, a non-central limit theorem is preserved in the case where the fluctuations of the estimator around the true value are non-Gaussian. As a direct consequence of convergence in \(L^2\), we obtain consistency of an estimator for the limiting self-similarity that does not require knowledge of the limiting diffusivity. Finally, we show that our results apply to a class of one-dimensional fluctuation models.