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

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,X 2Δn,...,X nΔn with sampling mesh Δn → 0 and the terminal sampling time nΔn → â̂ž. The rate of convergence turns out to be (aš nΔn, ǎš nΔn, ǎš n, ǎš 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. © 2012 EDP Sciences, SMAI.