Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records

TL;DR

This paper explores the universal statistical properties of the maximum and record counts in discrete-time random walks and Levy flights, revealing how these emerge from fundamental probabilistic theorems.

Contribution

It demonstrates the universal behavior of maximum and record statistics in random walks with arbitrary symmetric jump distributions, including Levy flights, based on Pollaczek-Spitzer and Sparre Andersen theorems.

Findings

Universal distribution of the global maximum and its occurrence time.

Universal statistics of record numbers and their ages.

Applicability to Levy flights and other symmetric jump distributions.

Abstract

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x_0=0, x_1,x_2.... x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Levy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.