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The paper is concerned with testing normality in samples of curves and error curves estimated from functional regression models. We propose a general paradigm based on the application of multivariate normality tests to vectors of functional principal components scores. We examine finite sample performance of a number of such tests and select the best performing tests. We apply them to several extensively used functional data sets and determine which can be treated as normal, possibly after a suitable transformation. We also offer practical guidance on software implementations of all tests we study and develop large sample justification for tests based on sample skewness and kurtosis of functional principal component scores.

We propose a broad class of models for time series of curves (functions) that can be used to quantify near long‐range dependence or near unit root behavior. We establish fundamental properties of these models and rates of consistency for the sample mean function and the sample covariance operator. The latter plays a role analogous to sample cross‐covariances for multivariate time series, but is far more important in the functional setting because its eigenfunctions are used in principal component analysis, which is a major tool in functional data analysis. It is used for dimension reduction of feature extraction. We also establish a central limit theorem for functions following our model. Both the consistency rates and the normalizations in the Central Limit Theorem (CLT) are nonstandard. They reflect the local unit root behavior and the long memory structure at moderate lags.