Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex

Let $d\in \mathbb{N}$ and let $γ_i\in [0,\infty)$, $x_i\in (0,1)$ be such that $\sum_{i=1}^{d+1} γ_i = M\in (0,\infty)$ and $\sum_{i=1}^{d+1} x_i = 1$. We prove that \begin{equation*} a \mapsto \frac{Γ(aM + 1)}{\prod_{i=1}^{d+1} Γ(a γ_i + 1)} \prod_{i=1}^{d+1} x_i^{aγ_i} \end{equation*} is completely monotonic on $(0,\infty)$. This result generalizes the one found by Alzer (2018) for binomial probabilities ($d=1$). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the $d$-dimensional simplex.

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