Some limit results on supremum of Shepp statistics for fractional Brownian motion

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• 作者：Zhong-quan Tan ; Yang Chen
• 关键词：Extremes ; Shepp statistics ; fractional Brownian motion ; exact asymptotic ; almost sure limit theorem
• 刊名：Applied Mathematics - A Journal of Chinese Universities
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：31
• 期：3
• 页码：269-282
• 全文大小：228 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Editorial Committee of Applied Mathematics - A Journal of Chinese Universities
• ISSN：1993-0445
• 卷排序：31

文摘

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.