Universal Record Statistics of Random Walks and L\'evy Flights

TL;DR

This paper demonstrates that record statistics for symmetric continuous random walks and Lévy flights are universal, independent of jump distribution details, with explicit growth laws for record counts and durations.

Contribution

It establishes the universality of record statistics for symmetric random walks and Lévy flights, providing explicit asymptotic growth formulas for record counts and durations.

Findings

Record mean grows as sqrt(4N/pi)

Standard deviation of records grows as sqrt((2-4/pi)N)

Shortest and longest record durations grow as sqrt(N/pi) and ~0.6265N

Abstract

It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution grows as the sqrt(4N/pi) while the standard deviation grows as sqrt((2-4/pi) N), so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt(N/pi) and 0.626508... N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.