Multiplicative processes as a source of fat-tail distributions.

Area of Science:

• Mathematical Physics
• Statistical Modeling

Background:

• The universality of power-law distributions is widely accepted in scientific literature.
• Recent research indicates more complex behaviors and exceptions to power-law universality.
• Conflicting findings necessitate a refined mathematical framework to understand power-law phenomena.

Purpose of the Study:

• To present two novel generalizations of power-law distributions.
• To reconcile the concept of power-law universality with emerging subtle details.
• To provide a mathematical framework for understanding diverse power-law generation mechanisms.

Main Methods:

• Developed a first generalization to analyze power-law generation via multiplicative processes.
• Introduced a second generalization involving the solution of Lambert's transcendental equations.
• Investigated the relationship between Pareto distributions and power multiplicative transformations.

Main Results:

• Demonstrated that a wide array of power-laws can arise from multiplicative processes, such as positive feedback.
• Showcased that the space of solutions where Pareto distributions match power multiplicative transformations is infinite.
• Provided analytical solutions for generalized power-law models.

Conclusions:

• The presented generalizations offer a more nuanced understanding of power-law distributions.
• Multiplicative processes are a key mechanism for generating diverse power-law behaviors.
• The infinite solution space highlights the complexity and adaptability of power-law models in various distributions.