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WEAK CONVERGENCE TO DERIVATIVES OF FRACTIONAL BROWNIAN MOTION

Published online by Cambridge University Press:  05 December 2022

Søren Johansen Open the ORCID record for Søren Johansen [Opens in a new window]  and
Morten Ørregaard Nielsen Open the ORCID record for Morten Ørregaard Nielsen [Opens in a new window]
Søren Johansen
Affiliation:
University of Copenhagen and CREATES
Morten Ørregaard Nielsen*
Affiliation:
Aarhus University
*
Address correspondence to Morten Ørregaard Nielsen, Department of Economics and Business Economics, Aarhus University, Aarhus, Denmark; e-mail: [email protected].
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Abstract

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It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter d converges weakly to fractional Brownian motion (fBm) for $d>\frac {1}{2}$. We show that, for any nonnegative 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 fBm. As an illustration, we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 4 , August 2024 , pp. 859 - 874
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to the Editor (Peter C.B. Phillips) and two anonymous referees for very helpful and detailed comments. We thank the Danish National Research Foundation for financial support (DNRF Chair Grant No. DNRF154).

References

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