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

A Gladyshev theorem for trifractional Brownian motion and $n$-th order fractional Brownian motion