Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion

TL;DR

This paper investigates the joint convergence of two subsequences of the signed cubic variation of fractional Brownian motion with Hurst parameter 1/6, showing they converge to a correlated two-dimensional Brownian motion with explicit covariance.

Contribution

It provides new conditions under which these subsequences converge jointly to a correlated two-dimensional Brownian motion with explicit covariance formulas.

Findings

Joint convergence to correlated 2D Brownian motion

Explicit covariance function formulas

Conditions on subsequences for convergence

Abstract

The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H=1/6$. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.