伯恩斯坦函数的应用：合并过程、Copula函数与从属关系
Bernstein Functions at Work: Coalescents, Copulas, and Subordination
AI总结：
研究随机过程等领域中若干正性问题，通过对完全单调函数等的识别演算及从属关系等方法，解决了如Möhle问题、Pearse - Bondell幂散度Copula生成器问题等。
摘要：
随机过程、相依性建模、分数分析和更新理论中的几个正性问题归结为一个共同的识别任务：归一化后，将对象识别为拉普拉斯变换、势密度、逆流系数或有限核平均，然后从该表示中读取符号模式。我们为完全单调函数、伯恩斯坦函数、特殊伯恩斯坦函数以及通过从属过程和混合测度的概率实现发展了这种识别演算。主要的肯定性结果解决了其源论文中使用的约定下的三个狭义源问题。通过有限单形有序对核证书证明了关于具有残余单元素质量（尘埃）的可交换合并过程的块计数过程的Möhle问题6.3。对于Pearse - Bondell幂散度Copula生成器，我们证明了在其第3.8节中确定的剩余严格负范围$\lambda\leq - 1$内逆的完全单调性。连同源论文中已经验证的特殊情况，这在每个维度上对于$\lambda\leq - 1$产生了阿基米德Copula。通过将势核表示为非增密度的伽马平均，回答了与特殊伯恩斯坦函数相关的离散更新序列的Bendikov - Cygan单调性问题。支持性的表示和边界结果涵盖了Sibisi的Prabhakar - Pollard $Q$ - 测度、Mecke - Nagel - Weiss在零处的原子以及立方分支准则$a^2\geq3b$。
英文摘要：
Several positivity questions in stochastic processes, dependence modeling, fractional analysis, and renewal theory reduce to a common recognition task: after normalization, identify the object as a Laplace transform, a potential density, an inverse-flow coefficient, or a finite kernel average, and then read the sign pattern from that representation. We develop this recognition calculus for completely monotone functions, Bernstein functions, special Bernstein functions, and probabilistic realizations through subordinators and mixing measures. The main affirmative results settle three narrowly stated source questions in the conventions used by their source papers. Möhle's Problem 6.3 on the block-counting process of exchangeable coalescents with residual singleton mass (dust) is proved by a finite-simplex ordered-pair kernel certificate. For the Pearse--Bondell power-divergence copula generators, we prove complete monotonicity of the inverse throughout the remaining strict negative range $λ\le-1$ identified in their Section 3.8. Together with the special cases already verified in the source paper, this yields Archimedean copulas in every dimension for $λ\le-1$. The Bendikov--Cygan monotonicity question for discrete renewal sequences attached to special Bernstein functions is answered by representing the potential kernel as a Gamma average of a nonincreasing density. Supporting representation and boundary results cover Sibisi's Prabhakar--Pollard $Q$-measure, the Mecke--Nagel--Weiss atom at zero, and the cubic branch criterion $a^2\ge3b$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
