On asymptotics of discretized functionals of long-range dependent functional data

Abstract

The paper studies the asymptotic behavior of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non central limit theorems for weighted additive functionals of random fields. As the majority of known results concern the discrete sampling case the developed methodology helps in translating these results to functional data without deriving them again. Numerical studies suggest that the theoretical findings are valid for wider classes of long-range dependent data.

Keywords:

• Additive functional
• functional data
• Hermite distributions
• long-range dependence
• non central limit theorems

Mathematics Subject Classification:

• 60G60
• 62M30
• 62M40
• 60F99

Acknowledgments

This research includes extensive simulation studies using the computational cluster Raijin of the National Computational Infrastructure (NCI), which is supported by the Australian Government and La Trobe University. The authors are also grateful to the referees for their suggestions that helped to improve the paper.

Additional information

Funding

This research was partially supported under the Australian Research Council’s Discovery Project DP160101366.