Long-Range Dependent Curve Time Series

Degui Li, Peter M. Robinson, Hanlin Shang

Research output: Contribution to journalArticlepeer-review


We introduce methods and theory for functional or curve time series with long-range dependence. The temporal sum of the curve process is shown to be asymptotically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run covariance function, which we use, via functional principal component analysis, in estimating the orthonormal functions spanning the dominant subspace of the curves. In a semiparametric context, we propose an estimate of the memory parameter and establish its consistency. A Monte Carlo study of finite-sample performance is included, along with two empirical applications. The first of these finds a degree of stability and persistence in intraday stock returns. The second finds similarity in the extent of long memory in incremental age-specific fertility rates across some developed nations. Supplementary materials for this article are available online.

Original languageEnglish
Pages (from-to)957-971
Number of pages15
JournalJournal of the American Statistical Association
Issue number530
Early online date30 May 2019
Publication statusPublished - Jun 2020

Bibliographical note

© 2019 American Statistical Association. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.


  • Curve process
  • Functional FARIMA
  • Functional principal component analysis
  • Limit theorems
  • Long-range dependence

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