Geodesic
Local asymptotic normality for normal inverse gaussian Lévy processes with high-frequency sampling
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 13-32

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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 Δ n , X 2 Δ n , ... , X n Δ n with sampling mesh Δ n → 0 and the terminal sampling time n Δ n → ∞ . The rate of convergence turns out to be ( n Δ n , n Δ 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.

DOI : 10.1051/ps/2011101
Classification : 60G51, 62E20
Keywords: high-frequency sampling, local asymptotic normality, normal inverse gaussian Lévy process 
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     author = {Kawai, Reiichiro and Masuda, Hiroki},
     title = {Local asymptotic normality for normal inverse gaussian {L\'evy} processes with high-frequency sampling},
     journal = {ESAIM: Probability and Statistics},
     pages = {13--32},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011101},
     mrnumber = {3002994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2011101/}
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Kawai, Reiichiro; Masuda, Hiroki. Local asymptotic normality for normal inverse gaussian Lévy processes with high-frequency sampling. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 13-32. doi: 10.1051/ps/2011101

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