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

TL;DR

This paper extends the big jump principle to correlated heavy-tailed random walks, showing how correlations influence the dominance of large jumps in extreme events and introducing two new principles for such systems.

Contribution

It introduces the big jump principle for correlated heavy-tailed random walks, analyzing how correlations affect the emergence of extreme events and proposing unconditioned and conditioned principles.

Findings

Correlation causes dependence of large sum values on subsequent increments.

Unconditioned big jump principle involves a correlation-dependent shift in distribution tails.

Conditional big jump principle depends on the step number of the big jump.

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 are the autoregressive model of first order and the discretized 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.