In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison and Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem is briefly mentioned but left open for future research.

中文翻译：

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狄利克雷密度的多元正态近似和一些应用

在这篇简短的笔记中，我们证明了 Dirichlet 密度与具有相同均值和协方差矩阵的多元正态密度之比的渐近展开。然后使用该展开式推导相应概率测度之间总变化的上限，并重新推导由 Aitchison 和 Lauder (1985) 引入并在 Ouimet (2020) 中进行理论研究的 Dirichlet 核估计量的渐近方差。简要提到了与高斯方差回归问题和高斯白噪声问题之间的渐近等价相关的另一个潜在应用，但留待未来研究。