Karhunen-Loeve expansions for the detrended Brownian motion

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• 作者：Xiaohui Ai^a ; ^{aixiaohui2006@gmail.com} ; Wenbo V. Li^b ; ^{wli@math.udel.edu} ; Guoqing Liu^a ; ^{liugq@hit.edu.com} ; ^{liugq@hit.edu.cn}
• 关键词：Detrended Brownian motion ; Karhunen&ndash ; Loeve expansions ; Laplace transform ; Large deviation ; Small deviation
• 刊名：Statistics and Probability Letters
• 出版年：2012
• 期刊代码：56_01677152
• 类别：mt
• 出版时间：July, 2012
• 卷：82
• 期：7
• 页码：1235-1241
• 文件大小：224 K

摘要

The detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the subspace spanned by linear functions. Karhunen-Loeve expansion for the process is obtained, together with the explicit formula for the Laplace transform of the squared norm. Distribution identities are established in connection with the second order Brownian bridge developed by . As applications, large and small deviation asymptotic behaviors for the norm are given.