Abstract
In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.
Funding Statement
This work was funded by CY Initiative of Excellence Paris Seine.
Acknowledgements
This project started while the author was an intern at Bloomberg. The author would like to thank Bruno Dupire and Sylvain Corlay for their valuable insights and discussions. The author is also grateful to A.B. Tsybakov for his comments on early drafts of this work. Finally, we thank the anonymous reviewers for their insightful comments.
Citation
Mohamed Ndaoud. "Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes." Bernoulli 29 (3) 2295 - 2317, August 2023. https://doi.org/10.3150/22-BEJ1542
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