  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.
