# Record statistics of a strongly correlated time series: random walks and L\'evy flights

## TL;DR

This paper reviews recent progress in understanding the statistics of record-breaking events in strongly correlated time series, especially random walks and Lévy flights, highlighting theoretical developments and applications.

## Contribution

It provides a comprehensive overview of record statistics in correlated processes, including new results for various random walk models and their physical implications.

## Key findings

- Record statistics differ significantly from i.i.d. cases due to correlations.
- Random walks with drift and constraints show unique record behaviors.
- Applications include physical systems with measurement noise.

## Abstract

We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a L\'evy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful "laboratory" to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statistics of the number of records, as well as of the ages of the records, i.e., the lapses of time between two successive record breaking events. Then we review the results that were obtained for a wide variety of random walk models, including random walks with a linear drift, continuous time random walks, constrained random walks (like the random walk bridge) and the case of multiple independent random walkers. Finally, we discuss further observables related to records, like the record increments, as well as some questions raised by physical applications of record statistics, like the effects of measurement error and noise.