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

TL;DR

This paper establishes limit theorems for the weighted quadratic variation of trifractional and n-th order fractional Brownian motions, providing tools for statistical estimation of their self-similarity indices.

Contribution

It extends existing results by proving new limit theorems and convergence conditions for these complex Gaussian processes.

Findings

Limit theorems for weighted quadratic variation of trifractional Brownian motion.

Limit theorems for weighted quadratic variation of n-th order fractional Brownian motion.

A statistical estimator for the self-similarity index of trifractional Brownian motion.

Abstract

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.