Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross-(auto)covariance matrix functions to accommodate more general sub-Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.
翻译:在这种假设下,除了功能数据的内在无限多元性外,功能变量的数量也会随着序列依赖性观测的数量而增加。在本文件中,我们侧重于对两个多变量功能时间序列之间或高原假设之外的多变量功能函数和星标时间序列混合体之间的相关估计交叉(自动)异差参数的理论分析。我们提出了功能跨光谱稳定度,以说明依赖这些估计的跨术语的影响,这些术语对于添加性功能线性回归估计数至关重要。在拟议功能跨光谱稳定度测量中,我们为估计的跨(自动)异差矩阵函数开发有用的集中性不平等,以适应更一般性的次加澳元函数线性线性进程,此外,在通常采用的功能性主要分析框架下,为相关估计术语建立有限的样本理论。我们用我们推算的不负线性结果,我们调查了两个添加性功能性线性回归应用的常规估计值的趋同性性质,这些假设包括功能性直线性轨/高度序列的功能性回归/部分直线性回归。