Descending from infinity: Convergence of tailed distributions

TL;DR

This paper studies how long-tailed distributions evolve under stochastic processes that tend to suppress their tails, revealing different relaxation regimes and transient behaviors.

Contribution

It introduces a detailed analysis of the relaxation dynamics of long-tailed distributions, identifying conditions for exponential suppression and transient phenomena.

Findings

Linear relaxation suppresses long tails exponentially over time.

Stronger relaxation causes immediate tail suppression but can produce transient peaks.

Two regimes of diffusive spreading are identified for certain initial conditions.

Abstract

We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not eliminated. Relaxation stronger than linear suppresses long tails immediately, but may lead to strong transient peaks in the probability distribution. A delta function initial distribution under stronger than linear decay displays not one but two different regimes of diffusive spreading.