Symmetric weighted odd-power variations of fractional Brownian motion and applications

TL;DR

This paper establishes a non-central limit theorem for symmetric weighted odd-power variations of fractional Brownian motion with Hurst parameter less than 1/2, and explores their asymptotic behavior in related stochastic processes.

Contribution

It introduces a non-central limit theorem for these variations and analyzes their asymptotics in fractional Brownian motion in Brownian time.

Findings

Proves a non-central limit theorem for H<1/2

Analyzes asymptotic behavior of trapezoidal weighted variations

Studies fractional Brownian motion in Brownian time

Abstract

We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H< 1/2. As applications, we study the asymptotic behavior of the trapezoidal weighted odd-power variations of the fractional Brownian motion and the fractional Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.