2 J an 2 00 9 Representation of gaussian small ball probabilities in l 2

Let z = P +∞ i=1 x2i /a 2 i where the xi’s are i.d.d centered with unit variance gaussian random variables and (ai)i∈N an increasing sequence such that P +∞ i=1 a i < +∞. We propose an exponential-integral representation theorem for the gaussian small ball probability P (z < ε) when ε ↓ 0. We start from a result by Meyer-Wolf, Zeitouni (1993) and Dembo, MeyerWolf, Zeitouni (1995) who computed this probability by means of series. We prove that P (z < ε) 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.