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.
