  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.
