  We give a two-dimensional central limit theorem (CLT) for the second-order
quadratic variation of the centered Gaussian processes on $[0,T]$. Though the
approach we use is well known in the literature, the conditions under which the
CLT holds are usually based on differentiability of the corresponding
covariance function. In our case, we replace differentiability conditions by
the convergence of the scaled sums of the second-order moments. To illustrate
the usefulness and easiness of use of the approach, we apply the obtained CLT
to proving the asymptotic normality of the estimator of the Orey index of a
subfractional Brownian motion.
