Estimating the complexity index of functional data: Some asymptotics

Consider a random curve valued in a general semi-metric space whose small-ball probability factorizes isolating the spatial and the volumetric term. Assume that the latter is specified and interprets its parameters as complexity indexes. An index estimate is constructed by comparing nonparametric versus parametric estimates of the volumetric factor, and various asymptotics (including weak convergence and asymptotic normality) are stated by means of U-statistics tools. As a by-product, new asymptotic results are stated for surrogate density estimation.