Uniform LAN property of locally stable L\'{e}vy process observed at high frequency

Suppose we have a high-frequency sample from the L\'{e}vy process of the form $X_t^\theta=\beta t+\gamma Z_t+U_t$, where $Z$ is a possibly asymmetric locally $\alpha$-stable L\'{e}vy process, and $U$ is a nuisance L\'{e}vy process less active than $Z$. We prove the LAN property about the explicit parameter $\theta=(\beta,\gamma)$ under very mild conditions without specific form of the L\'{e}vy measure of $Z$, thereby generalizing the LAN result of A\"{\i}t-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 $\alpha$-stable property, the asymptotic Fisher information matrix takes a clean-cut form.