Uniform LAN property of locally stable Lévy process observed at high frequency

Abstract

Suppose we have a high-frequency sample from the Lévy process of the form $X_t^θ=βt+γZ_t+U_t$, where $Z$ is a possibly asymmetric locally $α$-stable Lévy process, and $U$ is a nuisance Lévy process less active than $Z$. We prove the LAN property about the explicit parameter $θ=(β,γ)$ under very mild conditions without specific form of the Lévy measure of $Z$, thereby generalizing the LAN result of A\"ıt-Sahalia and Jacod (2007). In particular, it is clarified that a non-diagonal norming may be necessary in the truly asymmetric case. Due to the special nature of the local $α$-stable property, the asymptotic Fisher information matrix takes a clean-cut form.