Volume 17, 2013
|Page(s)||13 - 32|
|Published online||06 December 2012|
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.
2 Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.
Revised: 6 October 2010
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.
Mathematics Subject Classification: 60G51 / 62E20
Key words: High-frequency sampling / local asymptotic normality / normal inverse Gaussian Lévy process.
© EDP Sciences, SMAI, 2012
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