  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.
