Asymptotic theory for fractional regression models via Malliavin calculus

TL;DR

This paper investigates the asymptotic distribution of a sum involving fractional Brownian motions and kernel functions, revealing a mixed normal limit connected to the local time of the fractional Brownian motion, using Malliavin calculus techniques.

Contribution

It provides a new asymptotic theory for fractional regression models using Malliavin calculus, identifying mixed normal limits involving local times.

Findings

Limit distribution is a mixed normal law.

The limit involves the local time of fractional Brownian motion.

Results depend on the bandwidth parameter and Hurst indices.

Abstract

We study the asymptotic behavior as $n\to \infty$ of the sequence $$S_{n}=\sum_{i=0}^{n-1} K(n^{\alpha} B^{H_{1}}_{i}) (B^{H_{2}}_{i+1}-B^{H_{2}}_{i})$$ where $B^{H_{1}}$ and $B^{H_{2}}$ are two independent fractional Brownian motions, $K$ is a kernel function and the bandwidth parameter $\alpha$ satisfies certain hypotheses in terms of $H_{1}$ and $H_{2}$. Its limiting distribution is a mixed normal law involving the local time of the fractional Brownian motion $B^{H_{1}}$. We use the techniques of the Malliavin calculus with respect to the fractional Brownian motion.