The correction term in a Small--Ball Probability factorization for random curves

In this work we propose an analysis of the correction term appearing in a Small-Ball Probability factorization for random elements taking values in a separable Hilbert space. Its local nature, its meaning and behavior are discussed also through the derivation of some bounds. Nonparametric kernel--type estimators of the considered statistics are introduced and some asymptotic properties are provided. Finally, in the context of reconstructing a sample of curves by truncated Karhunen--Lo`{e}ve expansion, a local approach to select the dimensionality is illustrated through numerical and real data examples.