In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical stochastic processes with power-laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws such that the sums of the logarithmic return of non-identical stock price fluctuations follow stable distributions.

The exponential law has been discovered in various systems around the world. In this study, we introduce two existing and one proposed analytical method for exponential decay time-series predictions. The proposed method is given by a linear regression that is based on rescaling the time axis in terms of exponential decay laws. We confirm that the proposed method has a higher prediction accuracy than existing methods by performance evaluation using random numbers and verification using actual data. The proposed method can be used for analyzing real data modeled with exponential functions, which are ubiquitous in the world.

In recent years, the e-commerce market has grown with the spread of the internet worldwide every year. Accordingly, in service industries, purchasing products with reservations has become common. With the spread of online reservations, the booking curve, which is the concept of the time series in the cumulative number of reservations and has been used for sales optimization in the airline ticket and hotel industries, has been used in various industries. Booking curves in specific industries have been studied, but a universally applicable model across various industries has not been developed. In this study, we show that booking curves can be modeled universally by the exponential decay function, and we also show that the model is valid by using real data from some industries before and after the COVID-19 pandemic, that is, under completely different market conditions. The cross-industry exponential laws of booking curves constitute an important discovery in regard to mathematical laws in the social sciences and can be applied to give leading microeconomic indicators.

We propose a new dynamic pricing algorithm based on the universal exponential law of booking curves in services with reservations. The algorithm includes a parametric learning model which makes it possible to simulate the effect of changes in prices on quantity demanded from historical data continuously for practical use. Furthermore, we show an example, where some real data in a hotel applies for the learning model. Our proposed algorithm with the learning model, which can dynamically update the optimum parameters, is envisaged to be utilized as a practical dynamic pricing strategy.

The exponential law has been discovered in various systems around the world. In this study, we introduce two existing and one proposed analytical method for exponential decay time-series predictions. The proposed method is given by a linear regression that is based on rescaling the time axis in terms of exponential decay laws. We confirm that the proposed method has a higher prediction accuracy than existing methods by performance evaluation using random numbers and verification using actual data. The proposed method can be used for analyzing real data modeled with exponential functions, which are ubiquitous in the world.