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

Abstract

Let $X\sim \text{Mult}(N;p_1,\dots ,p_n)$ with $p_i>0$ and ${\sum }_i{p}_i=1$. We study $\mathrm{Pr}\{X=m\}$ 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 $m_i=\lfloor (N+d)p_i\rfloor$, $d\in [0,n)$. For $n\ge 3$ the mode need not be obtained by rounding each $N{p}_i$ to a neighbouring integer; the excess $m_i-\lceil N{p}_i\rceil$ can reach $n-2$, so the two-sided localisation $|m_i-N{p}_i|<1$ 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 $\mathrm{Pr}\{X=m\}$ 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 $\zeta$-value form.

21 pages