Yamashita Rios de Sousa Arthur Matsuo, Takayasu Hideki, Sornette Didier, Takayasu Misako
Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan.
Sony Computer Science Laboratories, Tokyo 141-0022, Japan.
Entropy (Basel). 2021 Feb 19;23(2):241. doi: 10.3390/e23020241.
The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.
本研究中所探讨的西格玛 - 派结构由数量不断增加的同分布随机变量的乘积之和构成。它出现在具有随机系数的随机过程中，也出现在诸如商业公司和城市等实体的增长模型中。我们研究具有伯努利随机变量的西格玛 - 派结构，发现无论随机变量是相互独立还是重复，其概率分布总是由一个幂律函数从下方界定。特别地，我们研究了渐近概率分布始终具有相同尾指数的上下幂律界的情况，该尾指数取决于随机变量分布的参数。我们在一个简单的增长模型背景下阐释西格玛 - 派结构，其中相继诞生的实体按照随机比例增长定律增长，分别采用伯努利随机变量（以证实理论结果）和半正态随机变量，对于半正态随机变量，其数值结果可利用伯努利情形的见解进行合理解释。我们分析了西格玛 - 派结构中由乘积项表示的实体之间的相互依赖性、增长因素中可能存在的记忆以及每个乘积项对整个西格玛 - 派结构的贡献。我们强调了实体之间相互依赖程度对有效贡献于规模总和的项数的影响，当所有实体独立增长时，达到单个项主导西格玛 - 派结构极值的极限情况。
