Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions

Abstract

We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We accommodate the framework of Jeganathan [Sankhyā Ser. A 44 (1982) 173–212] to a triangular array of variable dimension to, in particular, treat high-frequency observations of stochastic processes. When observations are smooth in the Malliavin sense, with the aid of Malliavin calculus techniques by Gobet [Bernoulli 7 (2001) 899–912], we further give tractable sufficient conditions which do not require Aronson-type estimates of the transition density function. The transition density function is even allowed to have zeros. For an application, we prove the local asymptotic mixed normality property of hypoelliptic diffusion models under high-frequency observations, in both complete and partial observation frameworks. The former and the latter extend previous results for elliptic diffusions and for integrated diffusions, respectively.