  We prove limit theorems for the weighted quadratic variation of trifractional
Brownian motion and $n$-th order fractional Brownian motion. Furthermore, a
sufficient condition for the $L^P$-convergence of the weighted quadratic
variation for Gaussian processes is obtained as a byproduct. As an application,
we give a statistical estimator for the self-similarity index of trifractional
Brownian motion. These theorems extend results of Baxter, Gladyshev, and
Norvai\v{s}a.
