A precise local limit theorem for the multinomial distribution and some applications.

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N − 1. In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R d. Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter (2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections. • Local limit theorem for the multinomial distribution up to an N − 3 ∕ 2 error. • Simple proof using Stirling's formula and Taylor expansions. • Application to obtain asymptotic properties of Bernstein estimators. • Improves Le Cam distance between multinomial and multivariate normal experiments. • Potential applications to continuity corrections and power divergence statistics. [ABSTRACT FROM AUTHOR]