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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 lessactive than $Z$. We prove the LAN property about the explicit parameter$\theta=(\beta,\gamma)$ under very mild conditions without specific form of theL\'{e}vy measure of $Z$, thereby generalizing the LAN result ofA\"{\i}t-Sahalia and Jacod (2007). In particular, it is clarified that anon-diagonal norming may be necessary in the truly asymmetric case. Due to thespecial nature of the local $\alpha$-stable property, the asymptotic Fisherinformation matrix takes a clean-cut form.

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