Effects of Randomness on Power Law Tails in Multiplicatively Interacting Stochastic Processes

TL;DR

This paper investigates how randomness affects the tail behavior of multiplicatively interacting stochastic processes, showing that randomness can decrease tail exponents and even reverse growth directions, especially under weak coupling and low growth conditions.

Contribution

It provides explicit calculations demonstrating the impact of randomness on tail exponents and growth rates in multiplicative stochastic processes, including cases where growth sign changes.

Findings

Randomness decreases tail exponents in stochastic processes.

Weak coupling amplifies the influence of randomness.

Growth rate sign can change from positive to negative due to randomness.

Abstract

Effects of randomness on non-integer power law tails in multiplicatively interacting stochastic processes are investigated theoretically. Generally, randomness causes decrease of the exponent of tails and the growth rate of processes. Explicit calculations are performed for two examples: uniformly distributed and two peaked systems. Significant influence is demonstrated when a bare growth rate is low and coupling is weak. It should be emphasized that even the sign of the growth rate can be changed from positive to negative growth.