Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Abstract

We consider statistical experiments associated with a Lévy process $X={\left(X_t\right)}_{t\ge 0}$ observed along a deterministic scheme $\left(i{u}_n,1\le i\le n\right)$ . We assume that under a probability ${\mathrm{\mathbb{P}}}_{\theta }$ , the r.v. $X_t,t>0$ , has a probability density function $>o$ , which is regular enough relative to a parameter $\theta \in \left(0,\infty \right)$ . We prove that the sequence of the associated statistical models has the LAN property at each $\theta$ , and we investigate the case when $X$ is the product of an unknown parameter $\theta$ by another Lévy process $Y$ with known characteristics. We illustrate the last results by the case where $Y$ is attracted by a stable process.