Conventionally, in data analysis used in statistics, the central limit theorem and the generalized central limit theorem have been used as basic statistical rules.However, in the case of price fluctuation distributions such as stock price fluctuations and exchange fluctuations in financial markets, probability statistical distributions such as the intervals at which earthquakes occur, and big data such as Internet traffic, the sum of rule distributions that should be different for each should be a basic form. The phenomenon of becoming a rule (stable distribution of Levi) can be seen.This could not be grasped by the conventional limit theorem, and the reason why such a phenomenon appeared was unknown.

　In this study, we formulated a statistical model called the sum of independent random variables that have different distributions that cannot be grasped by the conventional statistical rule, the limit theorem, which reflects actual data.Then, we derived the limit theorem that converges on the stable distribution of Levi in ​​the limit where the number N of data is infinite.

　This limit theorem is an extension of the generalized central limit theorem, which is a generalization of the central limit theorem, which is the basic law of statistics, to the limit of the sum of different power rules. Can be called.It has statistical significance as a limit theorem that holds even in more generalized situations, and as a basic statistical rule that accurately describes the characteristics of big data that show the universality of rules that should appear everywhere in the real world. It is also considered to have significance.

Paper information:[Journal of the Physical Society of Japan] Super Generalized Central Limit Theorem — Limit Distributions for Sums of Non-identical Random Variables with Power Laws —