Parameter estimation for fractional stochastic heat equations: Berry–Esseen bounds in CLTs

The aim of this work is to estimate the drift coefficient of a fractional heat equation driven by an additive space-time noise using the maximum likelihood estimator (MLE). In the first part of the paper, the first $N$ Fourier modes of the solution are observed continuously over a finite time interval $[0, T ]$. The explicit upper bounds for the Wasserstein distance for the central limit theorem (CLT) of the MLE are provided as $N \to \infty$ and/or $T \to \infty$. In the second part of the paper, the $N$ Fourier modes are observed at uniform time grid, $t_i = i (T/M)$, $i=0,\dots,M,$ where $M$ is the number of time grid points. The consistency and asymptotic normality are studied as $T,M,N \to +\infty$ in addition to the rate of convergence in law in the CLT.