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

In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting Fourier transform, where error terms are explicit up to order $N^{-1}$. this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and careful handling of Taylor expansions, show how result can be used approximate probabilities most subsets $\mathbb{R}^d$. Furthermore, discuss recent application obtain asymptotic properties Bernstein estimators simplex, improve main in Carter (2002) Le Cam distance bound between multivariate normal experiments while simultaneously simplifying proof, mention another potential related finely tuned continuity corrections.