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Path Properties of a Generalized Fractional Brownian Motion

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Abstract

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local law of the iterated logarithms.

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Acknowledgements

The authors are thankful to the reviewers and associate editor for their careful reading and suggestions on various improvements of the paper. In particular, the associate editor pointed out a gap in Sect. 4 in the paper, which led to a significant improvement of our understanding of the process. Tomoyuki Ichiba was supported in part by NSF Grants DMS-1615229 and DMS-2008427. Guodong Pang was supported in part by CMMI-1635410, DMS/CMMI-1715875 and the Army Research Office through Grant W911NF-17-1-0019. Murad S. Taqqu was supported in part by Simons Foundation Grant 569118 at Boston University.

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Authors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    Tomoyuki Ichiba

  2. The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, College of Engineering, Pennsylvania State University, University Park, PA, 16802, USA

    Guodong Pang

  3. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215, USA

    Murad S. Taqqu

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  1. Tomoyuki Ichiba
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  2. Guodong Pang
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  3. Murad S. Taqqu
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Correspondence to Guodong Pang.

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Ichiba, T., Pang, G. & Taqqu, M.S. Path Properties of a Generalized Fractional Brownian Motion. J Theor Probab 35, 550–574 (2022). https://doi.org/10.1007/s10959-020-01066-1

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  • DOI: https://doi.org/10.1007/s10959-020-01066-1

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