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Levy Flights, Autocorrelation, and Slow Convergence

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  • Sergio Da Silva

    (Federal University of Rio Grande Do Sul, Brazil)

Abstract

Previously we have put forward that the sluggish convergence of truncated Lévy flights to a Gaussian (Phys. Rev. Lett. 73 (1994) 2946) together with the scaling power laws in their probability of return to the origin (Nature 376 (1995) 46) can be explained by autocorrelation in data (Physica A 323 (2003) 601; Phys. Lett. A 315 (2003) 51). A purpose of this paper is to improve and enlarge the scope of such a result. The role of the autocorrelations in the convergence process as well as the problem of establishing the distance of a given distribution to the Gaussian are analyzed in greater detail. We show that whereas power laws in the second moment can still be explained by linear correlation of pairs, sluggish convergence can now emerge from nonlinear autocorrelations. Our approach is exemplified with data from the British pound–US dollar exchange rate.
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Suggested Citation

  • Sergio Da Silva, 2004. "Levy Flights, Autocorrelation, and Slow Convergence," Finance 0405021, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:0405021
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    References listed on IDEAS

    as
    1. Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2003. "Autocorrelation as a source of truncated Lévy flights in foreign exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 323(C), pages 601-625.
    2. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook9401.
    3. Peligrad, Magda & Shao, Qi-Man, 1995. "A note on the almost sure central limit theorem for weakly dependent random variables," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 131-136, February.
    4. Rafał Weron, 2001. "Levy-Stable Distributions Revisited: Tail Index> 2does Not Exclude The Levy-Stable Regime," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 209-223.
    5. repec:nys:sunysb:93-02 is not listed on IDEAS
    6. Matsuba, Ikuo & Takahashi, Hiroshi, 2003. "Generalized entropy approach to stable Lèvy distributions with financial application," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 319(C), pages 458-468.
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    Cited by:

    1. Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2006. "Nonidentically distributed variables and nonlinear autocorrelation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 171-180.
    2. Figueiredo, Annibal & Gleria, Iram & Matsushita, Raul & Da Silva, Sergio, 2005. "Financial volatility and independent and identically distributed variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 346(3), pages 484-498.
    3. Yang, Honglin & Wan, Hong & Zha, Yong, 2013. "Autocorrelation type, timescale and statistical property in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1681-1693.
    4. Matsushita, Raul & Figueiredo, Annibal & Da Silva, Sergio, 2012. "A suggested statistical test for measuring bivariate nonlinear dependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4891-4898.

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