Representation of small ball probabilities in Hilbert space and lower bound in regression for functional data

Let $S=\sum_{i=1}^{+\infty}\lambda_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\mathbb{E}\| Z\| ^{3}<+\infty$ and $(\lambda_{i})_{i\in\mathbb{N}}$ a positive nonincreasing sequence such that $\sum\lambda_{i}<+\infty$. We study the small ball probability $\mathbb{P}(S<\epsilon)$ when $\epsilon\downarrow0$. We start from a result by Lifshits (1997) who computed this probability by means of the Laplace transform of $S$. We prove that $\mathbb{P}(S<\cdot)$ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class of Gamma-varying functions, for which an exponential-integral representation is available. This approach allows to derive bounds for the rate in nonparametric regression for functional data at a fixed point $x_{0}$ : $\mathbb{E}(y|X=x_{0}$ where $(y_{i},X_{i})_{1\leq i\leq n}$ is a sample in $(\mathbb{R},\mathcal{F})$ and $\mathcal{F}$ is some space of functions. It turns out that, in a general framework, the minimax lower bound for the risk is of order $(\log n)^{-\tau}$ for some $\tau>0$ depending on the regularity of the data and polynomial rates cannot be achieved.