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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Research supported by National Science Foundation of China (11501250), Natural Science Foundation of Zhejiang Province of China (LQ14A010012, LY15A010019), Postdoctoral Research Program of Zhejiang Province, Natural Science Foundation of Jiangsu Higher Education Institution of China (14KJB110023) and Research Foundation of SUST.
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Tan, Zq., Chen, Y. Some limit results on supremum of Shepp statistics for fractional Brownian motion. Appl. Math. J. Chin. Univ. 31, 269–282 (2016). https://doi.org/10.1007/s11766-016-3417-9
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DOI: https://doi.org/10.1007/s11766-016-3417-9