调制分支过程、幂律的起源与混沌对偶
Modulated Branching Processes, Origins of Power Laws and Queueing Duality
摘要：
幂律分布在各种社会经济、生物和技术领域中被反复观察到。在许多观察中，例如，城市人口和生物体的大小，感兴趣的对象由于它们的许多独立成分的复制而进化，例如，个体的出生-死亡和细胞的复制。此外，复制的速率通常由引起扩张和收缩期的外源参数控制，例如，婴儿潮和婴儿潮、经济繁荣和经济衰退等。此外，这些对象的尺寸通常具有反射性的下边界，例如，城市不低于一定的规模，低收入个人由政府补贴，公司受破产法保护，等等。因此，很自然地提出反射调制分支过程作为许多前述观察的通用模型。事实上，我们的主要结果表明，所提出的数学模型的结果幂律分布在相当一般的多项式Gartner-Ellis条件下，其一般性可以解释无处不在的性质的幂律分布。此外，在对数尺度上，我们建立了反射分支过程和相应的乘法分支过程之间的渐近等价性。如Goldie（1991）所认识到的，后者已知是双晶化/加成过程。我们强调这种二重性进一步在平稳和遍历过程的一般性。
英文摘要：
Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991), is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes.
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