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 & 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.

翻译：在这个简短的注释中,我们证明,对于Drichlet密度与多变正常密度之比,我们用同样的中值和共差矩阵来进行无症状的扩大。然后,扩大用于得出相应的概率测量和重新校正Aitchison & Lauder(1985年)提出并在Oumet (202020年)中进行理论研究的Drichlet内核测深器的无症状差异之间的总差差值的上限。另一个潜在应用与Gaussian差异回归问题和Gaussian白噪音问题之间的无症状等值有关,对此简单提及,但留待今后研究。