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.

arXiv comment9 pages, 5 figures

References in corpus (21)

• The large deviation approach to statistical mechanics
• Lévy walks
• Stretched exponential distributions in Nature and Economy: ``Fat tails'' with characteristic scales
• On the strong anomalous diffusion
• Generalised extreme value statistics and sum of correlated variables
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• The Single Big Jump Principle in Physical Modelling
• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Lévy walks and scaling in quenched disordered media
• Constraint driven condensation in large fluctuations of linear statistics
• Infinite densities for Lévy walks
• Rare events in generalized Lévy Walks and the Big Jump principle
• Transport in disordered systems: the single big jump approach
• On the concentration of large deviations for fat tailed distributions, with application to financial data
• Superdiffusion and Transport in 2d-systems with Lévy Like Quenched Disorder
• Random walks in a one-dimensional Lévy random environment
• Mean squared displacement in a generalized Lévy walk model
• Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas
• Development and regression of a large fluctuation
• Continuous-time random walk between Lévy-spaced targets in the real line

Cited by in corpus (13)

• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Extreme value theory for constrained physical systems
• Exploring the Gillis model: a discrete approach to diffusion in logarithmic potentials
• Fast rare events in exit times distributions of jump processes
• Rare Events in Extreme Value Statistics of Jump Processes with Power Tails
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Large fluctuations and transport properties of the Lévy-Lorentz gas
• Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain
• Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes
• Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes
• Work fluctuations for a confined Brownian particle: the role of initial conditions
• Anatomy of an extreme event: What can we infer about the history of a heavy-tailed random walk?
• Big jump principle for heavy-tailed random walks with correlated increments