Abstract
We consider a non-ergodic class of stationary real harmonizable symmetric α-stable processes 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 with , 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
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
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