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Multinomial probabilities near the mode: integer modes and the complete local expansion
Let $X\sim\text{Mult}(N;p_1,\dots,p_n)$ with $p_i>0$ and $\sum_ip_i=1$. 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 u
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