  In Siotani & 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 $\mathbb{R}^d$.
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.
