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Extremes of Shepp statistics for fractional Brownian motion

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Abstract

Define the incremental fractional Brownian field with parameter H ∈ (0, 1) by Z H (τ, s) = B H (s +τ)-B H (s), where B H (s) is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We firstly derive the exact tail asymptotics for the maximum \(M_H^* (T) = \max _{(\tau ,s) \in [a,b] \times [0,T]} Z_H (\tau ,s)/\tau ^H \) of the standardised fractional Brownian motion field, with any fixed 0 < a < b < ∞ and T > 0; and we, furthermore, extend the obtained result to the case that T is a positive random variable independent of {B H (s), s ⩾ 0}. As a by-product, we obtain the Gumbel limit law for M * H (T) as T → ∞.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    ZhongQuan Tan

  2. School of Mathematics and Statistics, Nanjing Audit University, Nanjing, 211815, China

    Yang Yang

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  1. ZhongQuan Tan
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  2. Yang Yang
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Correspondence to ZhongQuan Tan.

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Tan, Z., Yang, Y. Extremes of Shepp statistics for fractional Brownian motion. Sci. China Math. 58, 1779–1794 (2015). https://doi.org/10.1007/s11425-014-4945-5

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  • DOI: https://doi.org/10.1007/s11425-014-4945-5

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