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

We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process X, when we observe high-frequency data XΔn ,X2Δn , . . . ,XnΔn with sampling mesh Δn→0 and the terminal sampling time nΔn→∞. The rate of convergence turns out to be (√nΔn,√nΔn,√n,√n) for the dominating parameter (α,β ,δ ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ 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.