This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

翻译：本文针对具有相同均值-协方差结构的负多项概率质量函数与多元正态密度之比的对数，提出了一种精炼的局部近似。该近似通过斯特林公式及对泰勒展开的细致处理推导得出，为抖动后的负多项分布与相应多元正态分布之间的Hellinger距离提供了上界。进而可得负多项实验与多元正态实验之间Le Cam距离的上界。