Record Statistics of Continuous Time Random Walk

TL;DR

This paper investigates record statistics in continuous time random walks, revealing independence from jump distribution details but dependence on waiting time distributions, with results expressed via Levy stable laws.

Contribution

It provides a detailed analysis of record statistics in CTRWs, deriving exact scaling functions and exploring age-related properties, highlighting the role of waiting time distributions.

Findings

Record statistics are independent of jump length distribution details.

The probability of M records follows a scaling form involving Levy stable laws.

Ages of records exhibit distinct asymptotic behaviors, with mean age <t/M> differing from t/<M>.

Abstract

The statistics of records for a time series generated by a continuous time random walk is studied, and found to be independent of the details of the jump length distribution, as long as the latter is continuous and symmetric. However, the statistics depend crucially on the nature of the waiting time distribution. The probability of finding M records within a given time duration t, for large t, has a scaling form, and the exact scaling function is obtained in terms of the one-sided Levy stable law. The mean of the ages of the records, defined as <t/M>, differs from t/<M>. The asymptotic behaviour of the shortest and the longest ages of the records are also studied.