Abstract
One of the main problem in prediction theory of discrete-time second-order stationary processes is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting given , , as n goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process . In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case – deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process and the differential properties of its spectral density f, while for deterministic processes it is determined by the geometric properties of the spectrum of and singularities of its spectral density f.
Funding Statement
The work is dedicated to our teacher Academician, Professor Il’dar Abdullovich Ibragimov on the occasion of his 90th birthday.
Acknowledgments
The authors are grateful to their supervisor Academician, Professor Il’dar Abdullovich Ibragimov for introducing them to this research area and permanent support.
The authors would like to thank the referees and Editor in Chief Mikhail Lifshits for their constructive comments and suggestions.
Citation
Nikolay M. Babayan. Mamikon S. Ginovyan. "Asymptotic behavior of the prediction error for stationary sequences." Probab. Surveys 20 664 - 721, 2023. https://doi.org/10.1214/23-PS21
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