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Ergodic properties of stationary stable processes

Author

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  • Cambanis, Stamatis
  • Hardin, Clyde D.
  • Weron, Aleksander

Abstract

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes, sufficient conditions are derived for metric transitivity and mixing, and necessary and sufficient conditions for a strong law of large numbers.

Suggested Citation

  • Cambanis, Stamatis & Hardin, Clyde D. & Weron, Aleksander, 1987. "Ergodic properties of stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 1-18, February.
    • Handle: RePEc:eee:spapps:v:24:y:1987:i:1:p:1-18
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    Citations

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    Cited by:

    1. Mathias Mørck Ljungdahl & Mark Podolskij, 2022. "Multidimensional parameter estimation of heavy‐tailed moving averages," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 593-624, June.
    2. Wang, Yizao & Stoev, Stilian A. & Roy, Parthanil, 2012. "Decomposability for stable processes," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1093-1109.
    3. Andreas Basse-O'Connor & Raphaël Lachièze-Rey & Mark Podolskij, 2015. "Limit theorems for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-56, Department of Economics and Business Economics, Aarhus University.
    4. Mathias Mørck Ljungdahl & Mark Podolskij, 2020. "A minimal contrast estimator for the linear fractional stable motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 381-413, July.
    5. Stilian Stoev & Murad S. Taqqu, 2005. "Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(2), pages 211-249, March.
    6. Andreas Basse-O'Connor & Mark Podolskij, 2015. "On critical cases in limit theory for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-57, Department of Economics and Business Economics, Aarhus University.
    7. Rybaczuk, M. & Weron, K., 1989. "Linearly coupled quantum oscillators with Lévy stable noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 160(3), pages 519-526.
    8. Hsing, Tailen, 1995. "Limit theorems for stable processes with application to spectral density estimation," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 39-71, May.
    9. Mazur, Stepan & Otryakhin, Dmitry & Podolskij, Mark, 2018. "Estimation of the linear fractional stable motion," Working Papers 2018:3, Örebro University, School of Business.
    10. Magdziarz, Marcin, 2009. "Correlation cascades, ergodic properties and long memory of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3416-3434, October.
    11. Kabluchko, Zakhar & Schlather, Martin, 2010. "Ergodic properties of max-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 281-295, March.
    12. Stoev, Stilian A., 2008. "On the ergodicity and mixing of max-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1679-1705, September.

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