Eur. Phys. J. B (2021) 94: 216
https://doi.org/10.1140/epjb/s10051-021-00215-7

Regular Article - Statistical and Nonlinear Physics

Big jump principle for heavy-tailed random walks with correlated increments

Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, 52900, Ramat Gan, Israel

Received: 29 June 2021
Accepted: 23 September 2021
Published online: 26 October 2021

Abstract

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into physical sciences where systems usually exhibit correlations. Here, we study the principle for a random walk with correlated increments. Examples of the increments are the autoregressive model of first order and the discretised Ornstein–Uhlenbeck process both with heavy-tailed noise. The correlation leads to the dependence of large values of the sum not only on the big jump but also on the following increments. We describe this behaviour by two big jump principles, namely unconditioned and conditioned on the step number when the big jump occurs. The unconditional big jump principle is described by a correlation-dependent shift between the sum and maximum distribution tails. For the conditional big jump principle, the shift depends also on the step number of the big jump.