Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

TL;DR

This paper analyzes the long-term behavior of continuous-time random walks with superheavy-tailed waiting times and heavy-tailed jumps, deriving their limiting distributions and providing analytical and numerical characterizations.

Contribution

It introduces a comprehensive framework for determining the scaling functions and limiting densities for such random walks with superheavy-tailed waiting times.

Findings

Derived explicit forms of limiting probability densities.

Established properties of the limiting densities.

Developed an efficient numerical method for density computation.

Abstract

We study the long-time behavior of the scaled walker (particle) position associated with decoupled continuous-time random walk which is characterized by superheavy-tailed distribution of waiting times and asymmetric heavy-tailed distribution of jump lengths. Both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution. To analytically investigate the limiting density function, we derive a number of different representations of this function and, by this way, establish its main properties. We also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results.