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Intermittency in the small-time behavior of Lévy processes

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  • Grahovac, Danijel

Abstract

In this paper we consider convergence of moments in the small-time limit theorems for Lévy processes. We provide precise asymptotics for all the absolute moments of positive order. The convergence of moments in limit theorems holds typically only up to some critical moment order and higher order moments decay at different rate. Such behavior is known as intermittency and has been encountered in some limit theorems.

Suggested Citation

  • Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:stapro:v:187:y:2022:i:c:s0167715222000852
    DOI: 10.1016/j.spl.2022.109507
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    References listed on IDEAS

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    1. Danijel Grahovac & Nikolai N. Leonenko & Murad S. Taqqu, 2020. "The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 33(4), pages 1801-1831, December.
    2. Grahovac, Danijel & Leonenko, Nikolai N. & Taqqu, Murad S., 2019. "Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5113-5150.
    3. R. A. Doney & R. A. Maller, 2002. "Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity," Journal of Theoretical Probability, Springer, vol. 15(3), pages 751-792, July.
    4. Grahovac, Danijel & Leonenko, Nikolai N. & Taqqu, Murad S., 2018. "Intermittency of trawl processes," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 235-242.
    5. J. F. Muzy & R. Baile & E. Bacry, 2013. "Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary $1/f$ noise. Application to volatility fluctuations in stock markets," Papers 1301.4160, arXiv.org.
    6. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
    7. Deng, Chang-Song & Schilling, René L., 2015. "On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3851-3878.
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