Optimal Berry-Esseen bound for statistical estimations and its application to SPDE

We consider asymptotically normal statistics of the form Fn/Gn, where Fn and Gn are functionals of Gaussian fields. For these statistics, we establish an optimal Berry-Esseen bound for the Central Limit Theorem (CLT) of the sequence Fn/Gn is φ(n) in the following sense: there exist constants 0