On raw moments of the binomial distribution

TL;DR

The paper analyzes the $k$-th raw moment $E(R^k)$ for $R \sim \mathrm{Binomial}(n,p)$ in the critical scaling where $n/k \to β>0$. It expresses the moment as a sum over Stirling numbers of the second kind and applies a refined saddle-point (Laplace) analysis, aided by Temme's expansion and unimodality arguments, to pinpoint an interior maximum that governs the sum. The main result is a precise asymptotic of the form $E(R^k) = k^k (Ψ + o(1))^k$, with the asymptotic factor $Ψ$ given explicitly in terms of $β$, $p$, and the principal Lambert W function. This yields the sharp asymptotic for the moment in the critical regime, improving over previous bounds and clarifying high-order moment behavior of binomial variables.

Abstract

We study the $k$:th raw moment of a variable $R$ following the binomial distribution $\text{B}(n, p)$, where $n/k \rightarrow β> 0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k Ψ^k$. We solve the asymptotically optimal value of $Ψ$ as a function of $p$ and $β$.

Figures (1)

• Figure 1: The isosceles triangle is inside the ziggurat.

Theorems & Definitions (4)

• Theorem 1
• Lemma 2: Klaner
• proof
• proof : Proof of Theorem \ref{['main']}.