Wergen Gregor, Majumdar Satya N, Schehr Grégory
Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jul;86(1 Pt 1):011119. doi: 10.1103/PhysRevE.86.011119. Epub 2012 Jul 18.
We study the statistics of the number of records R(n,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 σ(2) of the jump distribution is finite and (II) when σ(2) is divergent as in the case of Lévy flights with index 0<μ<2. In both cases we find that the mean record number R(n,N) grows universally as ~α(N) sqrt[n] for large n, but with a very different behavior of the amplitude α(N) for N>1 in the two cases. We find that for large N, α(N) ≈ 2sqrt[lnN] independently of σ(2) in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α(N) ≈ 4/sqrt[π], independently of 0<μ<2. For finite σ(2) we argue-and this is confirmed by our numerical simulations-that the full distribution of (R(n,N)/sqrt[n]-2sqrt[lnN])sqrt[lnN] converges to a Gumbel law as n → ∞ and N → ∞. In case II, our numerical simulations indicate that the distribution of R(n,N)/sqrt[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.
我们研究了一维中(N)个相同且独立的对称离散时间随机游走在(n)步时记录数(R(n,N))的统计特性，所有游走都在第(0)步从原点出发。在每个时间步，每个游走者以独立于对称连续分布抽取的随机长度跳跃。我们考虑两种情况：（I）跳跃分布的方差(\sigma^2)有限，以及（II）(\sigma^2)发散，如指数(0 < \mu < 2)的 Lévy 飞行情况。在这两种情况下，我们发现对于大的(n)，平均记录数(R(n,N))普遍以(\sim\alpha(N)\sqrt{n})的形式增长，但在两种情况下(N > 1)时幅度(\alpha(N))的行为非常不同。我们发现对于大的(N)，在情况 I 中(\alpha(N) \approx 2\sqrt{\ln N})，与(\sigma^2)无关。相比之下，在情况 II 中，对于大的(N)，幅度趋近于一个与(N)无关的常数，(\alpha(N) \approx 4 / \sqrt{\pi})，与(0 < \mu < 2)无关。对于有限的(\sigma^2)，我们论证（并且我们的数值模拟证实了这一点），当(n \to \infty)且(N \to \infty)时，((R(n,N)/\sqrt{n} - 2\sqrt{\ln N})\sqrt{\ln N})的完整分布收敛到 Gumbel 定律。在情况 II 中，我们的数值模拟表明，当(n \to \infty)且(N \to \infty)时，(R(n,N)/\sqrt{n})的分布收敛到一个与(\mu)无关的通用非平凡分布。我们讨论了我们的结果在研究标准普尔 500 指数的 366 个每日股票价格的记录统计中的应用。
