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

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 Lévy distributions, such as Lévy walks and the Lévy-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.

9 pages, 5 figures

References in corpus (6)

• The large deviation approach to statistical mechanics
• Lévy walks
• Infinite densities for Lévy walks
• Rare events in generalized Lévy Walks and the Big Jump principle
• Mean squared displacement in a generalized Lévy walk model
• Development and regression of a large fluctuation

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