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Can continuous-time stationary stable processes have discrete linear representations?

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  • Pipiras, Vladas
  • Taqqu, Murad S.
  • Abry, Patrice

Abstract

We show that a non-trivial continuous-time strictly [alpha]-stable, [alpha][set membership, variant](0,2), stationary process cannot be represented in distribution as a discrete linear processwhere is a collection of deterministic functions and are independent strictly [alpha]-stable random variables. Analogous results hold for self-similar strictly [alpha]-stable processes and for strictly [alpha]-stable processes with stationary increments. As a consequence, the usual wavelet decomposition of Gaussian self-similar processes cannot be extended to the [alpha]-stable, [alpha]

Suggested Citation

  • Pipiras, Vladas & Taqqu, Murad S. & Abry, Patrice, 2003. "Can continuous-time stationary stable processes have discrete linear representations?," Statistics & Probability Letters, Elsevier, vol. 64(2), pages 147-157, August.
    • Handle: RePEc:eee:stapro:v:64:y:2003:i:2:p:147-157
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    References listed on IDEAS

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    1. Hardin, Clyde D., 1982. "On the spectral representation of symmetric stable processes," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 385-401, September.
    2. Cambanis, Stamatis & Maejima, Makoto, 1989. "Two classes of self-similar stable processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 305-329, August.
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    1. Soltani, A.R. & Parvardeh, A., 2005. "Decomposition of discrete time periodically correlated and multivariate stationary symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1838-1859, November.
    2. Grassi, Stefano & Santucci de Magistris, Paolo, 2014. "When long memory meets the Kalman filter: A comparative study," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 301-319.
    3. Matsui, Muneya & Takemura, Akimichi, 2009. "Integral representations of one-dimensional projections for multivariate stable densities," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 334-344, March.

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