  We study rates of convergence in central limit theorems for the partial sum
of squares of general Gaussian sequences, using tools from analysis on Wiener
space. No assumption of stationarity, asymptotically or otherwise, is made. The
main theoretical tool is the so-called Optimal Fourth Moment Theorem
\cite{NP2015}, which provides a sharp quantitative estimate of the total
variation distance on Wiener chaos to the normal law. The only assumptions made
on the sequence are the existence of an asymptotic variance, that a
least-squares-type estimator for this variance parameter has a bias and a
variance which can be controlled, and that the sequence's auto-correlation
function, which may exhibit long memory, has a no-worse memory than that of
fractional Brownian motion with Hurst parameter }$H<3/4$.{\ \ Our main result
is explicit, exhibiting the trade-off between bias, variance, and memory. We
apply our result to study drift parameter estimation problems for subfractional
Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes with
fixed-time-step observations. These are processes which fail to be stationary
or self-similar, but for which detailed calculations result in explicit
formulas for the estimators' asymptotic normality.
