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

Published online by Cambridge University Press:  06 December 2012

Reiichiro Kawai  and
Hiroki Masuda
Reiichiro Kawai
Affiliation:
School of Mathematics and Statistics, University of Sydney NSW 2006, Australia.. reiichiro.kawai@maths.usyd.edu.au
Hiroki Masuda
Affiliation:
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.; hiroki@imi.kyushu-u.ac.jp
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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 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.

Keywords

High-frequency samplinglocal asymptotic normalitynormal inverse Gaussian Lévy process.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 13 - 32
Copyright
© EDP Sciences, SMAI, 2012

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