Multipower variation from generalized difference for fractional integral processes with jumps

ABSTRACT

This article presents limit theorems of the multipower variation based on a generalized difference for the fractional integral process with jumps observed in high frequency. In particular, we obtain the large number laws for threshold multipower variation and multipower variation and the associated central limit theorems. The limit theorems are applied to estimate Hurst parameter, and the consistence and asymptotic distribution of the estimator are established. These results will provide some new statistical tools to analyze long-memory effect in high-frequency situation.

KEYWORDS:

• Central limit theorem
• Fractional integral process
• High-frequency data
• Higher-order differences
• Multipower variation.

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F05
• 60F17
• 60G22

Acknowledgments

The authors are thankful to the editor-in-chief, Prof. N. Balakrishnan, the editorial assistant Ms. Dobbie Iscoe, and a referee, for their help and the valuable suggestions which helped to improve this article significantly.

Funding

The work was partially supported by National NSFC (No. 11501503), Natural Science Foundation of Jiangsu Province of China (No. BK20131340), China Postdoctoral Science Foundation (No. 2014M560471, 2016T90534), QingLan Project of Jiangsu Province of China, Priority Academic Program Development of Jiangsu Higher Education Institutions (Applied Economics), and Key Laboratory of Jiangsu Province (Financial Engineering Laboratory).