Weak convergence to derivatives of fractional Brownian motion

TL;DR

This paper proves that derivatives of normalized fractional processes with respect to the fractional parameter converge weakly to derivatives of fractional Brownian motion, extending known convergence results and applying them to multifractional models.

Contribution

It establishes joint weak convergence of derivatives of fractional processes to derivatives of fractional Brownian motion for any non-negative integer order.

Findings

Derivatives of fractional processes converge weakly to derivatives of fractional Brownian motion.

Results apply to asymptotic distributions in multifractional vector autoregressive models.

Extends classical convergence results to derivatives with respect to the fractional parameter.

Abstract

It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$, derivatives of order $m=0,1,\dots,M$ of the normalized fractional process with respect to the fractional parameter $d$, jointly converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.