Record statistics for multiple random walks

We study the statistics of the number of records R-n,R-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 sigma(2) of the jump distribution is finite and (II) when sigma(2) is divergent as in the case of Levy flights with index 0 < mu < 2. In both cases we find that the mean record number < R-n,R-N > grows universally as similar to alpha(N) root n for large n, but with a very different behavior of the amplitude alpha(N) for N > 1 in the two cases. We find that for large N, alpha(N) approximate to 2 root ln N independently of sigma(2) in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, alpha(N) approximate to 4/root pi, independently of 0 < mu < 2. For finite sigma(2) we argue-and this is confirmed by our numerical simulations-that the full distribution of (R-n,R-N/root n - 2 root ln N)root ln N converges to a Gumbel law as n -> infinity and N -> infinity. In case II, our numerical simulations indicate that the distribution of R-n,R-N/root n converges, for n -> infinity and N -> infinity, to a universal nontrivial distribution independently of mu. 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.