Representation of gaussian small ball probabilities in $l_2$

André Mas (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let $z=\sum_{i=1}^{+\infty}x_{i}^{2}/a_{i}^{2}$ where the $x_{i}$'s are i.d.d centered with unit variance gaussian random variables and $\left( a_{i}\right) _{i\in\mathbb{N}}$ an increasing sequence such that $\sum _{i=1}^{+\infty}a_{i}^{-2}<+\infty$. We propose an exponential-integral representation theorem for the gaussian small ball probability $\mathbb{P}% \left( z<\varepsilon\right) $ when $\varepsilon\downarrow0$. We start from a result by Meyer-Wolf, Zeitouni (1993) and Dembo, Meyer-Wolf, Zeitouni (1995) who computed this probability by means of series. We prove that $\mathbb{P}% \left( z<\varepsilon\right) $ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class Gamma, for which an explicit exponential-integral representation is available. The converse implication holds under a mild additional assumption. Some applications are underlined in connection with statistical inference for random functions.

Mots clés: de Haan's Gamma class; small ball problems; regular variations; gaussian random elements

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

André Mas. Representation of gaussian small ball probabilities in $l_2$. 2008. ⟨hal-00349575v1⟩