Some limit results on supremum of Shepp statistics for fractional Brownian motion |
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Authors: | Zhong-quan Tan Yang Chen |
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Institution: | 1.School of Mathematical Sciences,Zhejiang University,Hangzhou,China;2.College of Mathematics, Physics and Information Engineering,Jiaxing University,Jiaxing,China;3.School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou,China |
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Abstract: | Define the incremental fractional Brownian field Z H (τ, s) = B H (s + τ) ? B H (s), where B H (s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). In this paper, we first derive an exact asymptotic of distribution of the maximum \({M_H}\left( {{T_u}} \right) = {\sup _{r \in 0,1],s \in 0,x{T_u}]}}{Z_H}\left( {\tau ,s} \right)\), which holds uniformly for x ∈ A,B] with A,B two positive constants. We apply the findings to analyse the tail asymptotic and limit theorem of M H (T) with a random index T. In the end, we also prove an almost sure limit theorem for the maximum M 1/2(T) with non-random index T. |
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