Nonparametric inference of a trend using functional data

Let X = {X(t), t ∈ [0, T ]} be a second order random process of which n independent realizations are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function μ = EX in the space C([0, T ]) as n, p → ∞ and, using Gaussian process theory, we derive approximate simultaneous confidence bands for μ.