We study sufficient conditions for a local asymptotic mixed normality property of statistical models. We develop a scheme with the $L^2$ regularity condition proposed by Jeganathan [\textit{Sankhya Ser. A} \textbf{44} (1982) 173--212] so that it is applicable to high-frequency observations of stochastic processes. Moreover, by combining with Malliavin calculus techniques by Gobet [\textit{Bernoulli} \textbf{7} (2001) 899--912, 2001], we introduce tractable sufficient conditions for smooth observations in the Malliavin sense, which do not require Aronson-type estimates of the transition density function. Our results, unlike those in the literature, can be applied even when the transition density function has zeros. For an application, we show the local asymptotic mixed normality property of degenerate (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.

翻译：我们为统计模型的局部零和混合常态特性研究足够的条件。 我们开发了一个由Jeganathan[\ textit{ Sankhya Ser. A}\ textbf{44}(1982)173-212) 提议的以美元为单位的常规性条件($L2$)的方案,以便适用于对随机过程的高频观测。此外,我们通过Gobet[\ text{Bernoul}\ textbf{7}(2001)899-912,2001]与Malliavin 技术的Malliavin 技术结合,我们引入了在马利亚文意义上的平稳观测所需的充足条件,不需要对过渡密度函数进行Aronson型的估算。我们的结果,与文献中的结果不同,即使在过渡密度函数为零时,也可以应用。对于应用,我们展示了高频观测框架下的堕落(超电离子)传播模型的局部无常态混合性常态特性,包括完整和部分观测框架。前一个和后一个观测框架分别扩展了前一个和综合传播结果。