二项分布关于给定中心的绝对偏差：闭式尾部展开
Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form
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中文总结 AI 辅助
本文针对二项分布关于给定中心的绝对偏差，给出止损尾部展开的闭式系数，结合伯努利多项式、贝尔递推和欧拉多项式，并揭示有效展开参数与格点缺陷。
中文摘要 AI 辅助
设 $X\sim Bin(N,p)$，并设 $c=Nr$ 为给定中心，其中 $r\ne p$。$E|X-c|$ 的有限公式是经典的，而完整的 $N^{-1}$ 幂次大偏差展开的存在性由 Timashev 对二项尾部的鞍点展开得出。我们给出绝对偏差产生的止损尾部的闭式系数，并显式保留 $Nr$ 的格点缺陷。该构造结合三个标准要素：用于连续二项质量的伯努利多项式、指数贝尔递推以及用于加权几何和的欧拉多项式。第 $k$ 个系数的分母恰好为 $(1-\rho)^{2k+2}$，其中 $\rho$ 是指数倾斜；因此有效展开参数为 $1/(N(1-\rho)^2)$，这是不完全伽马展开中参数的格点加权类比。$r=\tfrac12$ 的情形给出有偏 $\pm1$ 随机游走在原点的折叠，其奇偶分裂被识别为折叠的格点缺陷。
英文摘要
Let $X\sim Bin(N,p)$ and let $c=Nr$ be a prescribed centre with $r\ne p$. The finite formula for $E|X-c|$ is classical, and the existence of a complete large-deviation expansion in powers of $N^{-1}$ follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of $Nr$ retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The $k$-th coefficient has denominator exactly $(1-ρ)^{2k+2}$, where $ρ$ is the exponential tilt; hence the effective expansion parameter is $1/(N(1-ρ)^2)$, the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case $r=\tfrac12$ gives the fold at the origin of a biased $\pm1$ random walk, and its parity split is identified with the lattice defect of the fold.
发表机构
- University of Zagreb（萨格勒布大学）
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