Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$

TL;DR

This paper analyzes the asymptotic behavior of weighted quadratic variations of fractional Brownian motion specifically at the critical Hurst index $H=1/4$, completing recent research and solving a conjecture on Riemann sums.

Contribution

It provides the asymptotic analysis for the critical case $H=1/4$, filling a gap in existing literature and addressing a recent conjecture.

Findings

Derived the asymptotic behavior for $H=1/4$ case.

Solved a conjecture on Riemann sums with alternating signs.

Completes the analysis for all Hurst indices in this context.

Abstract

We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.