通过 Mills 比的 Stein 解因子：仿射出生率与对称势分布

中文摘要 AI 辅助

我们通过一种在离散和连续情形下通用的 Mills 比方法，研究指示函数检验函数的 Stein 解因子。在离散情形下，一般的生灭过程表述给出了精确的解包络，并恢复了 Brown 和 Xia 的一阶增量理论。对于二项分布和泊松分布目标，以及形状参数 $r\ge1$ 的负二项分布目标，仿射出生率的额外代数结构改进了 Stein 解的均匀界。在连续情形下，我们考虑与 $e^{-V}$ 成比例的对称密度。在关于势 $V$ 的自然结构假设下，Mills 比为指示函数 Stein 解提供了显式的有限界，并表明均匀导数和漂移 Stein 因子具有精确值 $1$。在额外的一个交叉条件下，解因子优化可以精确执行，并给出最优值 $1/(4p(0))$。统计力学中出现的偶次幂目标，包括四次临界 Curie--Weiss 定律，提供了原始例子，且该计算扩展到更广泛的 Subbotin 族。对于 Subbotin 族中的 $1<\beta<2$，精确包络具有两个由唯一的不完全伽马方程刻画的偏离中心的极大值点，而两个一阶因子仍等于 $1$。这些结果展示了改进的，以及在连续尖锐情形下最优的 Stein 解因子背后的一个共同的离散-连续机制。

英文摘要

We study Stein solution factors for indicator test functions by a common Mills-ratio method in discrete and continuous settings. In the discrete case, a general birth--death formulation gives exact solution envelopes and recovers the first-increment theory of Brown and Xia. For binomial and Poisson targets, and for negative binomial targets with shape parameter $r\ge1$, the additional algebraic structure of affine birth rates yields improved uniform bounds for the Stein solution. In the continuous case, we consider symmetric densities proportional to $e^{-V}$. Under natural structural assumptions on the potential $V$, Mills ratios yield explicit finite bounds for the indicator Stein solution and show that the uniform derivative and drift Stein factors have the sharp value $1$. Under an additional one-crossing condition, the solution-factor optimization can be carried out exactly and gives the optimal value $1/(4p(0))$. The even-power targets arising in statistical mechanics, including the quartic critical Curie--Weiss law, provide the original examples, and the calculation extends to the wider Subbotin family. For $1<β<2$ in the Subbotin family, the exact envelope has two off-center maximizers characterized by a unique incomplete-gamma equation, while the two first-order factors remain equal to $1$. The results exhibit a common discrete--continuous mechanism behind improved and, in the continuous sharp regime, optimal Stein solution factors.

发表机构

• Faculty of Mathematics, Ruhr University Bochum（波鸿鲁尔大学数学学院）

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