On the likelihood function of small time variance Gamma Lévy processes

Abstract

We investigate the likelihood function of small generalized Laplace laws and variance gamma Lévy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Lévy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Lévy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.

Keywords:

• fisher information
• high-frequency sampling
• generalized laplace distribution
• Laplace Lévy motion
• local asymptotic normality

2010 Mathematics Subject Classification:

• 62F12
• 62E20
• 60E07
• 60G51

The author is grateful to three anonymous referees for very careful reading and constructive suggestions. Part of this work was carried out while the author was based at University of Leicester, UK.