Local asymptotic normality for normal inverse gaussian Lévy processes with high-frequency sampling

Kawai, Reiichiro; Masuda, Hiroki

ESAIM: Probability and Statistics, Tome 17 (2013), pp. 13-32.

We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process $X$, when we observe high-frequency data $X_{{\Delta }_n},X_{2{\Delta }_n},...,X_{n{\Delta }_n}$ with sampling mesh ${\Delta }_n\to 0$ and the terminal sampling time $n{\Delta }_n\to \infty$. The rate of convergence turns out to be $(\sqrt{n}{\Delta }_n,\sqrt{n}{\Delta }_n,\sqrt{n},\sqrt{n})$ for the dominating parameter $(\alpha ,\beta ,\delta ,\mu )$, where $\alpha$ stands for the heaviness of the tails, $\beta$ the degree of skewness, $\delta$ the scale, and $\mu$ the location. The essential feature in our study is that the suitably normalized increments of $X$ in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

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