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

¹ School of Mathematics and Statistics, University of Sydney NSW 2006, Australia.
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² Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.
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Received: 26 April 2010
Revised: 6 October 2010

Abstract

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_Δn,X_2Δn,...,X_nΔn with sampling mesh Δ_n → 0 and the terminal sampling time nΔ_n → ∞. The rate of convergence turns out to be (√nΔ_n, √nΔ_n, √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.