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Lower bound for local oscillations of Hermite processes

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  • Ayache, Antoine

Abstract

The most known example of a class of non-Gaussian stochastic processes which belongs to the homogeneous Wiener chaos of an arbitrary order N>1 is probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound for the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N, which, among other things, shows that these paths are nowhere differentiable functions.

Suggested Citation

  • Ayache, Antoine, 2020. "Lower bound for local oscillations of Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4593-4607.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4593-4607
    DOI: 10.1016/j.spa.2020.01.009
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    References listed on IDEAS

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    1. Maejima, Makoto & Tudor, Ciprian A., 2013. "On the distribution of the Rosenblatt process," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1490-1495.
    2. Ayache, Antoine & Roueff, François & Xiao, Yimin, 2009. "Linear fractional stable sheets: Wavelet expansion and sample path properties," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1168-1197, April.
    3. Bai, Shuyang & Taqqu, Murad S., 2015. "Convergence of long-memory discrete kth order Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 2026-2053.
    4. Bai, Shuyang & Taqqu, Murad S., 2014. "Generalized Hermite processes, discrete chaos and limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1710-1739.
    5. Bai, Shuyang & Taqqu, Murad S., 2014. "Structure of the third moment of the generalized Rosenblatt distribution," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 144-152.
    6. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
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    Cited by:

    1. Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.
    2. Ayache, Antoine & Bouly, Florent, 2022. "Moving average Multifractional Processes with Random Exponent: Lower bounds for local oscillations," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 143-163.
    3. Kerchev, George & Nourdin, Ivan & Saksman, Eero & Viitasaari, Lauri, 2021. "Local times and sample path properties of the Rosenblatt process," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 498-522.

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