Universal Record Statistics of Random Walks and Lévy Flights
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 4N/π while the standard deviation grows as (2-4/π)N, so the distribution is non-self-averaging. The mean shortest and longest duration records grow as N/π and 0.626508…N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.
- Publication:
-
Physical Review Letters
- Pub Date:
- August 2008
- DOI:
- 10.1103/PhysRevLett.101.050601
- arXiv:
- arXiv:0806.0057
- Bibcode:
- 2008PhRvL.101e0601M
- Keywords:
-
- 05.40.Fb;
- 02.50.Cw;
- 02.50.Sk;
- 24.60.-k;
- Random walks and Levy flights;
- Probability theory;
- Multivariate analysis;
- Statistical theory and fluctuations;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 4 pages, 3 figures. Added journal ref. and made small changes. Compatible with published version