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

Abramowitz, 1964, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55, xiv+1046 Arenbaev, 1976, Asymptotic behavior of the multinomial distribution, Teor. Veroyatn. Primen., 21, 826 Babu, 2002, Application of Bernstein polynomials for smooth estimation of a distribution and density function, J. Statist. Plann. Inference, 105, 377, 10.1016/S0378-3758(01)00265-8 Babu, 2006, Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors, Statist. Probab. Lett., 76, 959, 10.1016/j.spl.2005.10.031 Belalia, 2016, On the asymptotic properties of the Bernstein estimator of the multivariate distribution function, Statist. Probab. Lett., 110, 249, 10.1016/j.spl.2015.10.004 Bhattacharya, 1976, xiv+274 Bikyalis, 1969, Asymptotic expansions for distributions of sums of independent equally distributed lattice random variables, Theory Probab. Appl., 14, 481, 10.1137/1114060 Carter, 2002, Deficiency distance between multinomial and multivariate normal experiments. Dedicated to the memory of Lucien Le Cam, Ann. Statist., 30, 708, 10.1214/aos/1028674839 Chacón, 2018, Multivariate kernel smoothing and its applications, vol. 160, xxi+226 Cressie, 1978, A finely tuned continuity correction, Ann. Inst. Statist. Math., 30, 435, 10.1007/BF02480234 Kolassa, 1994, Series approximation methods in statistics, vol. 88, viii+150 Lange, 1995, Applications of the Dirichlet distribution to forensic match probabilities, Genetica, 96, 107, 10.1007/BF01441156 Lazakovičius, 1969, Asymptotic expansions for distributions of sums of independent lattice random vectors, Lith. Math. J., 22, 45, 10.1007/BF00967926 Le Cam, 2000, Asymptotics in statistics, xiv+285 Leblanc, 2012, On estimating distribution functions using Bernstein polynomials, Ann. Inst. Statist. Math., 64, 919, 10.1007/s10463-011-0339-4 Leblanc, 2012, On the boundary properties of Bernstein polynomial estimators of density and distribution functions, J. Statist. Plann. Inference, 142, 2762, 10.1016/j.jspi.2012.03.016 Ledoux, 2001, The concentration of measure phenomenon, no. 89, x+181 Mariucci, 2016, Le Cam theory on the comparison of statistical models., Grad. J. Math., 1, 81 Mosimann, 1962, On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions, Biometrika, 49, 65 Ouimet, 2018, Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex, J. Math. Anal. Appl., 466, 1609, 10.1016/j.jmaa.2018.06.049 Ouimet, 2019, 425 Ouimet, 2020, 1 Ouimet, 2020, 1 Ouimet, 2020, 1 Ouimet, 2021, General formulas for the central and non-central moments of the multinomial distribution, Stats, 4, 18, 10.3390/stats4010002 Ouimet, 2021, On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators, J. Math. Anal. Appl., 499, 10.1016/j.jmaa.2021.125033 Prokhorov, 1953, Asymptotic behavior of the binomial distribution, Usp. Mat. Nauk., 8, 135 Prokhorov, 2013, Some approximation problems in statistics and probability, vol. 42, 235 Read, 1984, Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics, Ann. Inst. Statist. Math., 36, 59, 10.1007/BF02481953 Scott, 2015, Multivariate density estimation, xviii+350 Severini, 2005, Elements of distribution theory, vol. 17, xii+515 Siotani, 1984, Asymptotic approximations for the distributions of multinomial goodness-of-fit statistics, Hiroshima Math. J., 14, 115, 10.32917/hmj/1206133150 Steele, 1997, Probability theory and combinatorial optimization, vol. 69, viii+159 Tanabe, 1992, An exact Cholesky decomposition and the generalized inverse of the variance-covariance matrix of the multinomial distribution, with applications, J. R. Stat. Soc. Ser. B Stat. Methodol., 54, 211 Tenbusch, 1994, Two-dimensional Bernstein polynomial density estimators, Metrika, 41, 233, 10.1007/BF01895321 Ulyanov, 2009, Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution, Hiroshima Math. J., 39, 133, 10.32917/hmj/1237392382 Vitale, 1975, Bernstein polynomial approach to density function estimation, 87