Gregor Wergen

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length \eta drawn from a continuous distribution f(\eta) which is symmetric around a constant drift c. We focus in particular on the case were f(\eta) is a symmetric stable law with a L\'evy index 0 < \mu \leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter \mu. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and \mu) of the jump distribution. We compute the asymptotic mean record number after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < \mu \leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

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 $\sigma${}2 of the jump distribution is finite and (II) when $\sigma${}2 is divergent as in the case of L\'evy flights with index 0<$\mu${}1 in the two cases. We find that for large N, $\alpha${}N$\approx${}2lnN 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$\approx${}4/$\pi${}, independently of 0<$\mu${}<2. For finite $\sigma${}2 we argue\char22{}and this is confirmed by our numerical simulations\char22{}that the full distribution of (Rn,N/n$-${}2lnN)lnN converges to a Gumbel law as n$\rightarrow${}$\infty${} and N$\rightarrow${}$\infty${}. In case II, our numerical simulations indicate that the distribution of Rn,N/n converges, for n$\rightarrow${}$\infty${} and N$\rightarrow${}$\infty${}, 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.

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 \sigma^2 of the jump distribution is finite and (II) when \sigma^2 is divergent as in the case of L\'evy flights with index 0 < \mu < 2. In both cases we find that the mean record number grows universally as \sim \alpha_N \sqrt{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 \approx 2 \sqrt{\log 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 \approx 4/\sqrt{\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,N}/\sqrt{n} - 2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N \to \infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty, 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 & Poors 500 index.