Rare Events in Extreme Value Statistics of Jump Processes with Power Tails

Abstract

We study rare events in the extreme value statistics of stochastic symmetric jump processes with power tails in the distributions of the jumps, using the big-jump principle. The principle states that in the presence of stochastic processes with power tails statistics, if at a certain time a physical quantity takes on a value much larger than its typical value, this large fluctuation is realised through a single macroscopic jump that exceeds the typical scale of the process by several orders of magnitude. In particular, our estimation focuses on the asymptotic behaviour of the tail of the probability distribution of maxima, a fundamental quantity in a wide class of stochastic models used in chemistry to estimate reaction thresholds, in climatology for earthquake risk assessment, in finance for portfolio management, and in ecology for the collective behaviour of species. We determine the analytical form of the probability distribution of rare events in the extreme value statistics of three jump processes with power tails; Lévy flights, Lévy walks and the Lévy-Lorentz gas. For the Lévy flights, we re-obtain through the big-jump approach recent analytical results, extending their validity. For the Lévy-Lorentz gas we show that the topology of the disordered lattice along which the walker moves induces memory effects in its dynamics, which influences the extreme value statistics. Our results are confirmed by extensive numerical simulations.

9 pages, 6 figures

References in corpus (6)

• Lévy walks
• Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution
• Discrete Sampling of Extreme Events Modifies Their Statistics
• Scaling properties of field-induced superdiffusion in Continous Time Random Walks
• Fast rare events in exit times distributions of jump processes
• Extreme value statistics of positive recurrent centrally biased random walks

Cited by in corpus (5)

• Optimal conditions for first passage of jump processes with resetting
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes
• Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain
• Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables