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Nonparametric estimation of the mean and covariance functions is ubiquitousin functional data analysis and local linear smoothing techniques are mostfrequently used. Zhang and Wang (2016) explored different types of asymptoticproperties of the estimation, which reveal interesting phase transitionphenomena based on the relative order of the average sampling frequency persubject $T$ to the number of subjects $n$, partitioning the data into threecategories: ``sparse'', ``semi-dense'' and ``ultra-dense''. In an increasinglyavailable high-dimensional scenario, where the number of functional variables$p$ is large in relation to $n$, we revisit this open problem from anon-asymptotic perspective by deriving comprehensive concentration inequalitiesfor the local linear smoothers. Besides being of interest by themselves, ournon-asymptotic results lead to elementwise maximum rates of $L_2$ convergenceand uniform convergence serving as a fundamentally important tool for furtherconvergence analysis when $p$ grows exponentially with $n$ and possibly $T$.With the presence of extra $\log p$ terms to account for the high-dimensionaleffect, we then investigate the scaled phase transitions and the correspondingelementwise maximum rates from sparse to semi-dense to ultra-dense functionaldata in high dimensions. Finally, numerical studies are carried out to confirmour established theoretical properties.

From sparse to dense functional data in high dimensions: Revisiting phase transitions from a non-asymptotic perspective