Rare events in generalized L\'evy Walks and the Big Jump principle

TL;DR

This paper analyzes rare events in generalized Lévy walks, demonstrating that large fluctuations are caused by single extreme steps, with derived probability tails revealing non-universal behaviors and offering insights for prediction.

Contribution

It extends the Big Jump principle to generalized Lévy walks, deriving exact tail distributions and revealing non-universal, non-analytic behaviors based on single-step dynamics.

Findings

Tail distributions depend on single-step dynamics

Rare events are driven by single large fluctuations

Provides a physical explanation for rare event mechanisms

Abstract

The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized L\'evy walks, a class of stochastic processes with power law distributed step durations, which model complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.