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 $X_t^{\theta }=\beta t+\gamma Z_t+U_t$, where $Z$ is a possibly asymmetric locally $\alpha$-stable Lévy process, and $U$ is a nuisance Lévy 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 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 $\alpha$-stable property, the asymptotic Fisher information matrix takes a clean-cut form.

24 pages

Cited by in corpus (2)

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