  Modelling a large bundle of curves arises in a broad spectrum of real
applications. However, existing literature relies primarily on the critical
assumption of independent curve observations. In this paper, we provide a
general theory for large-scale Gaussian curve time series, where the temporal
and cross-sectional dependence across multiple curve observations exist and the
number of functional variables, $p,$ may be large relative to the number of
observations, $n.$ We propose a novel functional stability measure for
multivariate stationary processes based on their spectral properties and use it
to establish some useful concentration bounds on the sample covariance matrix
function. These concentration bounds serve as a fundamental tool for further
theoretical analysis, in particular, for deriving nonasymptotic upper bounds on
the errors of the regularized estimates in high dimensional settings. As {\it
functional principle component analysis} (FPCA) is one of the key techniques to
handle functional data, we also investigate the concentration properties of the
relevant estimated terms under a FPCA framework. To illustrate with an
important application, we consider {\it vector functional autoregressive
models} and develop a regularization approach to estimate autoregressive
coefficient functions under the sparsity constraint. Using our derived
nonasymptotic results, we investigate the theoretical properties of the
regularized estimate in a "large $p,$ small $n$" regime. The finite sample
performance of the proposed method is examined through simulation studies.
