We prove the local asymptotic normality for the full parameters of the normal inverseGaussian Lévy process X, when we observe high-frequency data XΔn,X2Δn,...,XnΔnwith sampling mesh Δn → 0 and the terminalsampling time nΔn → ∞. Therate of convergence turns out to be (√nΔn, √nΔn, √n, √n) for the dominating parameter(α,β,δ,μ), where α stands for the heaviness of thetails, β the degree of skewness, δ the scale, andμ the location. The essential feature in our study is that the suitablynormalized increments of X in small time is approximatelyCauchy-distributed, which specifically comes out in the form of the asymptotic Fisherinformation matrix.

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.