Multinomial probabilities near the mode: integer modes and the complete local expansion

Abstract

Let X∼Mult(N;p₁,…,p_n) with p_i>0 and ∑_ip_i=1. We study Pr{X=m} for lattice points near Np from two complementary viewpoints. First, we give an exact integer-mode criterion: the multinomial mode is the Jefferson--D'Hondt divisor apportionment, given when the mode is unique by m_i=⌊ (N+d)p_i⌋, d∈[0,n). For n≥3 the mode need not be obtained by rounding each Np_i to a neighbouring integer; the excess m_i−⌈ Np_i⌉ can reach n−2, so the two-sided localisation |m_i−Np_i|<1 stated in a standard reference is not valid in general. Second, by writing the mass as a gamma quotient with unequal scalings, we derive the complete local expansion of Pr{X=m} in integer powers of N⁻¹, with all coefficients in closed Bernoulli-polynomial form. The expansion contains the known O(N⁻¹) local limit theorem, reduces to the binomial local mass for n=2 and to the central multinomial coefficient in the symmetric case, and admits Cesàro averages of the oscillating coefficients in closed ζ-value form.

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BibTeX RIS

BibTeX

@misc{elezovic2026multinomial,
  title = {{Multinomial probabilities near the mode: integer modes and the complete local expansion}},
  author = {Neven Elezović},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20229},
  eprint = {2609.20229},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20229},
}

RIS

TY  - UNPB
TI  - Multinomial probabilities near the mode: integer modes and the complete local expansion
AU  - Neven Elezović
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20229
UR  - https://arxiv.org/abs/2609.20229
AB  - Let X∼Mult(N;p_1,…,p_n) with p_i>0 and ∑_ip_i=1. We study Pr{X=m} for lattice points near Np from two complementary viewpoints. First, we give an exact integer-mode criterion: the multinomial mode is the Jefferson--D'Hondt divisor apportionment, given when the mode is unique by m_i=⌊ (N+d)p_i⌋, d∈[0,n). For n≥3 the mode need not be obtained by rounding each Np_i to a neighbouring integer; the excess m_i−⌈ Np_i⌉ can reach n−2, so the two-sided localisation |m_i−Np_i|<1 stated in a standard reference is not valid in general. Second, by writing the mass as a gamma quotient with unequal scalings, we derive the complete local expansion of Pr{X=m} in integer powers of N^−1, with all coefficients in closed Bernoulli-polynomial form. The expansion contains the known O(N^−1) local limit theorem, reduces to the binomial local mass for n=2 and to the central multinomial coefficient in the symmetric case, and admits Cesàro averages of the oscillating coefficients in closed ζ-value form.
ER  -

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