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Correlation cascades, ergodic properties and long memory of infinitely divisible processes

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  • Magdziarz, Marcin

Abstract

In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divisible processes in the language of the correlation cascade and establish its relationship with the codifference. Using the correlation cascade, we investigate the dependence structure of four fractional [alpha]-stable stationary processes. We detect the property of long memory and verify the ergodic properties of the discussed processes.

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  • 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.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3416-3434
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    References listed on IDEAS

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    1. Eliazar, Iddo & Klafter, Joseph, 2007. "Correlation cascades of Lévy-driven random processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 1-26.
    2. 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.
    3. Krzysztof Burnecki & Makoto Maejima & Aleksander Weron, 1997. "The Lamperti transformation for self-similar processes," HSC Research Reports HSC/97/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    4. Rosinski, Jan & Zak, Tomasz, 1996. "Simple conditions for mixing of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 277-288, February.
    5. Gross, Aaron, 1994. "Some mixing conditions for stationary symmetric stable stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 51(2), pages 277-295, July.
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