Record statistics of integrated random walks and the random acceleration process

TL;DR

This paper analyzes the record statistics of integrated random walks and the random acceleration process, deriving universal analytical expressions for record durations in the continuum limit.

Contribution

It provides the first analytical results for record durations in the random acceleration process, revealing their universality and connection to integrated random walks.

Findings

Derived the distribution of total record run durations

Established universality of results independent of step distribution details

Connected discrete integrated walks to the continuum random acceleration process

Abstract

We address the theory of records for integrated random walks with finite variance. The long-time continuum limit of these walks is a non-Markov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated random walks. This result is universal, i.e., independent of the details of the parent distribution of the step lengths.