Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle

TL;DR

This paper reviews and extends the big jump principle for rare events in stochastic processes with heavy-tailed distributions, applying it to continuous time random walks and Lorentz gas models with stretched exponential distributions.

Contribution

It generalizes the rate formalism for rare events to new models, providing analytic probability density functions for these processes.

Findings

Derived analytic PDFs for rare events in CTRWs and Lorentz gas models.

Extended the big jump principle to processes with stretched exponential distributions.

Clarified properties of stretched exponential distributions in rare event contexts.

Abstract

Rare events in stochastic processes with heavy-tailed distributions are controlled by the big jump principle, which states that a rare large fluctuation is produced by a single event and not by an accumulation of coherent small deviations. The principle has been rigorously proved for sums of independent and identically distributed random variables and it has recently been extended to more complex stochastic processes involving L\'evy distributions, such as L\'evy walks and the L\'evy-Lorentz gas, using an effective rate approach. We review the general rate formalism and we extend its applicability to continuous time random walks and to the Lorentz gas, both with stretched exponential distributions, further enlarging its applicability. We derive an analytic form for the probability density functions for rare events in the two models, which clarify specific properties of stretched exponentials.