Universal record statistics of random walks and Lévy flights

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 sqrt[4N/pi] while the standard deviation grows as sqrt[(2-4/pi)N], so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt[N/pi] and 0.626 508...N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.