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

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 $H\in (3\mathrm{/}4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in (1\mathrm{/}2,3\mathrm{/}4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3\mathrm{/}4$, whereas for the other values, that is, $H\in (1\mathrm{/}2,3\mathrm{/}4)$, the central convergence does not take place. We also provide Berry–Esseen estimates for the estimator.