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Local times and sample path properties of the Rosenblatt process

Author

Listed:
  • Kerchev, George
  • Nourdin, Ivan
  • Saksman, Eero
  • Viitasaari, Lauri

Abstract

Let Z=(Zt)t≥0 be the Rosenblatt process with Hurst index H∈(1∕2,1). We prove joint continuity for the local time of Z, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of Z. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in the second chaos.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:498-522
    DOI: 10.1016/j.spa.2020.09.018
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    References listed on IDEAS

    as
    1. Ayache, Antoine, 2020. "Lower bound for local oscillations of Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4593-4607.
    2. Nourdin, Ivan & Diu Tran, T.T., 2019. "Statistical inference for Vasicek-type model driven by Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3774-3791.
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