February 2025 Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes
Ly Viet Hoang, Evgeny Spodarev
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Bernoulli 31(1): 162-186 (February 2025). DOI: 10.3150/24-BEJ1723

Abstract

We consider a non-ergodic class of stationary real harmonizable symmetric α-stable processes X=X(t):tR with a finite symmetric and absolutely continuous control measure. We refer to its density function as the spectral density of X. These processes admit a LePage series representation and are conditionally Gaussian, which allows us to derive the non-ergodic limit of sample functions on X. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of X and the lag process X(t+h)X(t):tR with h>0, respectively. The process admits an equivalent representation as a series of sinusoidal waves with random frequencies which are i.i.d. with the (normalized) spectral density of X as their probability density function. Based on strongly consistent frequency estimation using the periodogram we present a strongly consistent estimator of the spectral density. The computation of the periodogram is fast and efficient, and our method is not affected by the non-ergodicity of X.

Acknowledgments

The authors would like to thank the editors and referees for their valuable feedback, which greatly improved the quality of this manuscript.

Citation

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Ly Viet Hoang. Evgeny Spodarev. "Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes." Bernoulli 31 (1) 162 - 186, February 2025. https://doi.org/10.3150/24-BEJ1723

Information

Received: 1 September 2022; Published: February 2025
First available in Project Euclid: 30 October 2024

Digital Object Identifier: 10.3150/24-BEJ1723

Keywords: Fourier analysis , frequency estimation , harmonizable process , non-ergodic process , non-ergodic statistics , periodogram , spectral density estimation , Stable process , stationary process

Vol.31 • No. 1 • February 2025
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