发表机构
FH Aachen - University of Applied Sciences; Universidad Autónoma de Madrid（亚琛应用科学大学; 马德里自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Banach空间中的局部平稳时间序列建立了函数中心极限定理，不依赖type-2或cotype假设，并应用于自归一化CUSUM变点检测。
AI 中文摘要
本文建立了取值于可分Banach空间$B$的局部平稳时间序列的函数中心极限定理。该结果不要求type-2或cotype假设，因此涵盖了函数型数据分析中的核心空间，包括$C([0,1])$和$L^p([0,1])$。在矩条件、可和物理依赖系数以及控制无穷维尾部的括号熵条件下，中心化和重标度的部分和过程在$D([0,1],B)$中弱收敛。极限是一个中心化的$B$值高斯过程，其协方差由局部长期协方差的积分给出，并解释为射影张量积的元素。我们还获得了关于柱状布朗运动的随机积分表示，将Banach空间极限与熟悉的标量局部平稳结构联系起来。作为应用，我们推导了一个自归一化CUSUM程序，用于检测Banach值观测的线性投影均值的变点，得到渐近枢轴零分布。通过蒙特卡洛实验以及对EEG记录和每日温度曲线的探索性应用，说明了有限样本行为。基于$C([0,1])$中Faber-Schauder系统和$W^{1,p}_0([0,1])$中$p$-Laplacian模型的例子，展示了如何验证熵条件。
英文摘要
A functional central limit theorem for locally stationary time series taking values in a separable Banach space $B$ is established. The result does not require type-2 or cotype assumptions and therefore covers spaces central to functional data analysis, including $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients, and a bracketing entropy condition controlling the infinite-dimensional tails, the centered and rescaled partial sum process converges weakly in $D([0,1],B)$. The limit is a centered $B$-valued Gaussian process whose covariance is given by the integral of the local long-run covariance, interpreted as an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion, connecting the Banach-space limit to the familiar scalar locally stationary structure. As an application, we derive a self-normalized CUSUM procedure for detecting changes in the mean of linear projections of Banach-valued observations, yielding a pivotal asymptotic null distribution. The finite-sample behavior is illustrated through Monte Carlo experiments and exploratory applications to EEG recordings and daily temperature curves. Examples based on the Faber-Schauder system in $C([0,1])$ and on a $p$-Laplacian model in $W^{1,p}_0([0,1])$ demonstrate how the entropy condition can be verified.
CommentsKeywords: Locally stationary time series; Banach-valued time series; functional central limit theorem; physical dependence; functional data analysis
