We are interested in the rate function of the moderate deviation principle for the two-sample matching problem. This is related to the determination of 1-Lipschitzfunctions with maximal variance. We give an exact solution for random variables which havenormal law, or are uniformly distributed on the Euclidean ball.

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by twoparameters. The first parameter governs the lacunarity of the waveletcoefficients while the second one governs its intensity. In this paper,we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal (in the Le Cam sense) tests on the intensity parameter.

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_t^{-}:= \int_0^t 1_{X_s < 0}{\rm d}s, t\geq 0)$ . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).