AI 中文总结
本文针对四种经典离散分布，证明其平均绝对偏差的坍缩恒等式，推导相关渐近展开并揭示级数包络特性，扩展了配套论文的二项分布展开。
AI 中文摘要
对于二项分布、泊松分布、负二项分布和超几何分布这四种经典离散律，其关于均值的平均绝对偏差会坍缩为单个点质量。我们给出这些恒等式的一个通用望远镜证明，并通过大小偏差解释所得闭式。随后，我们推导了泊松分布（λ→∞）、负二项分布（r→∞，p固定）和超几何分布（N→∞，边际成固定比例）的完全渐近展开，扩展了配套论文中的二项分布展开。系数以伯努利多项式闭式形式给出，精确携带均值的格位移；在整数均值处，这些展开简化为符号交替的奇级数，且单个Binet核论证表明，这些级数包络归一化平均绝对偏差的对数，即连续部分和对其进行夹逼。
英文摘要
For each of the four classical discrete laws --- binomial, Poisson, negative binomial and hypergeometric --- the mean absolute deviation about the mean collapses to a single point mass. We give a common telescoping proof of these identities and interpret the resulting closed forms by size biasing. We then derive complete asymptotic expansions for the Poisson ($λ\to\infty$), negative binomial ($r\to\infty$, $p$ fixed) and hypergeometric ($N\to\infty$, margins in fixed proportion) cases, extending the binomial expansion from the companion papers. The coefficients are given in closed Bernoulli-polynomial form and carry the lattice displacement of the mean exactly. At integer means the expansions reduce to sign-alternating odd series, and a single Binet-kernel argument shows that these series envelop the logarithm of the normalised mean absolute deviation: successive partial sums bracket it.
Comments23 pages
