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We study sufficient conditions for a local asymptotic mixed normalityproperty of statistical models. We develop a scheme with the $L^2$ regularitycondition proposed by Jeganathan [\textit{Sankhya Ser. A} \textbf{44} (1982)173--212] so that it is applicable to high-frequency observations of stochasticprocesses. Moreover, by combining with Malliavin calculus techniques by Gobet[\textit{Bernoulli} \textbf{7} (2001) 899--912, 2001], we introduce tractablesufficient conditions for smooth observations in the Malliavin sense, which donot require Aronson-type estimates of the transition density function. Ourresults, unlike those in the literature, can be applied even when thetransition density function has zeros. For an application, we show the localasymptotic mixed normality property of degenerate (hypoelliptic) diffusionmodels under high-frequency observations, in both complete and partialobservation frameworks. The former and the latter extend previous results forelliptic diffusions and for integrated diffusions, respectively.

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