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

Abstract

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.

21 pages, 0 figure, v4: minor corrections

References in corpus (5)

• Asymptotic properties of Bernstein estimators on the simplex
• General formulas for the central and non-central moments of the multinomial distribution
• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• On the boundary properties of Bernstein estimators on the simplex
• Explicit formulas for the joint third and fourth central moments of the multinomial distribution

Cited by in corpus (12)

• Asymptotic properties of Dirichlet kernel density estimators
• Asymptotic properties of Bernstein estimators on the simplex
• Quantum State Tomography for Matrix Product Density Operators
• General formulas for the central and non-central moments of the multinomial distribution
• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• On the boundary properties of Bernstein estimators on the simplex
• A multivariate normal approximation for the Dirichlet density and some applications
• A refined continuity correction for the negative binomial distribution and asymptotics of the median
• Refined normal approximations for the central and noncentral chi-square distributions and some applications
• On the Le Cam distance between multivariate hypergeometric and multivariate normal experiments
• Selecting a number of voters for a voting ensemble
• Explicit formulas for the joint third and fourth central moments of the multinomial distribution