Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

We consider statistical experiments associated with a Lévy process X = X t t ≥ 0 observed along a deterministic scheme i u n , 1 ≤ i ≤ n . We assume that under a probability ℙ θ , the r.v. X t , t > 0 , has a probability density function > o , which is regular enough relative to a parameter θ ∈ 0 , ∞ . We prove that the sequence of the associated statistical models has the LAN property at each θ , and we investigate the case when X is the product of an unknown parameter θ by another Lévy process Y with known characteristics. We illustrate the last results by the case where Y is attracted by a stable process.