Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian--fractional Brownian model

TL;DR

This paper investigates the asymptotic normality of a randomized periodogram estimator for quadratic variation in a mixed Brownian-fractional Brownian model, identifying conditions for CLT validity and providing Berry-Esseen bounds.

Contribution

It establishes the asymptotic distribution of the estimator across different Hurst parameter regimes and offers quantitative convergence rates.

Findings

Central limit theorem holds for H in (3/4,1)

Normal convergence with nonzero mean at H=3/4

No CLT for H in (1/2,3/4)

Abstract

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator.