Record statistics for multiple random walks
Abstract
We study the statistics of the number of records Rn,N for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ2 of the jump distribution is finite and (II) when σ2 is divergent as in the case of Lévy flights with index 0<μ<2. In both cases we find that the mean record number <Rn,N> grows universally as ∼αNn for large n, but with a very different behavior of the amplitude αN for N>1 in the two cases. We find that for large N, αN≈2lnN independently of σ2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, αN≈4/π, independently of 0<μ<2. For finite σ2 we argue—and this is confirmed by our numerical simulations—that the full distribution of (Rn,N/n-2lnN)lnN converges to a Gumbel law as n→∞ and N→∞. In case II, our numerical simulations indicate that the distribution of Rn,N/n converges, for n→∞ and N→∞, to a universal nontrivial distribution independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poor's 500 index.
- Publication:
-
Physical Review E
- Pub Date:
- July 2012
- DOI:
- 10.1103/PhysRevE.86.011119
- arXiv:
- arXiv:1204.5039
- Bibcode:
- 2012PhRvE..86a1119W
- Keywords:
-
- 05.40.-a;
- 02.50.Sk;
- Fluctuation phenomena random processes noise and Brownian motion;
- Multivariate analysis;
- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Disordered Systems and Neural Networks;
- Mathematics - Probability;
- Physics - Data Analysis;
- Statistics and Probability;
- Quantitative Finance - Statistical Finance
- E-Print:
- 25 pages, 8 figures