Central Limit Theorems and Minimum-Contrast Estimators for Linear Stochastic Evolution Equations

TL;DR

This paper establishes central limit theorems and asymptotic properties for minimum-contrast estimators of the drift in linear stochastic evolution equations driven by fractional Brownian motion, covering both singular and regular cases.

Contribution

It introduces new limit theorems and asymptotic normality results for estimators in fractional stochastic evolution equations using the 4th moment theorem on Wiener chaos.

Findings

Proves strong consistency via ergodicity of stationary solutions.

Establishes asymptotic normality for Hurst parameter H < 3/4.

Provides Berry-Esseen bounds for convergence speed.

Abstract

Central limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ($H < \frac{1}{2})$ and regular ($H > \frac{1}{2})$ types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter $H < \frac{3}{4}$) is the so-called $4^{th}$ moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.