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 η drawn from a continuous distribution f (η) which is symmetric around a constant drift c. We focus in particular on the case were f (η) is a symmetric stable law with a Lévy index 0 < µ ≤ 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 µ. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and µ) of the jump distribution. We compute the asymptotic mean record number R n 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 < µ ≤ 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

Abstract. -We present a mathematical analysis of records drawn from independent random variables with a drifting mean. To leading order the change in the record rate is proportional to the ratio of the drift velocity to the standard deviation of the underlying distribution. We apply the theory to time series of daily temperatures for given calendar days, obtained from historical climate recordings of European and American weather stations as well as re-analysis data. We conclude that the change in the mean temperature has increased the rate of record breaking events in a moderate but significant way: For the European station data covering the time period 1976-2005, we find that about 5 of the 17 high temperature records observed on average in 2005 can be attributed to the warming climate.Introduction. -In current media coverage the occurrence of record-breaking temperatures and other extreme weather conditions is often associated with global climate change. However, record breaking events occur at a certain rate in any stationary random process. In mathematical terms, a record is an entry in a time series that is larger (upper record) or smaller (lower record) than all previous entries [1][2][3]. If the entries are independent and identically distributed random variables drawn from a continuous probability distribution, the probability P n to observe a new record after n steps, hereafter referred to as the record rate, is simply P n = 1/n, because all n values are equally likely to be the largest. Applying this result to maximal temperatures measured at a specific calendar day over a time span of n years, it follows that the expected number of records per year is 365/n, i.e. about 12 records for an observation period of 30 years. Remarkably, this prediction is entirely independent of the underlying probability distribution, which may even differ for different calendar days.Despite considerable current interest in extreme climate events [4][5][6][7][8][9][10][11][12][13][14], the subject of climate records has received relatively little attention. It is intuitively obvious that an increase in the mean temperature will lead to an increased occurrence of high temperature records, but attempts to detect this effect in observational data have long remained inconclusive [15][16][17][18]. Only very recently an empirical study of temperature data from the US found a significant effect of warming on the relative occurrence of

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.

Abstract. We consider records and sequences of records drawn from discrete time series of the form X n = Y n + cn, where the Y n are independent and identically distributed random variables and c is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability p n (c) of a record occurring in the nth step and the probability P N (c) that all N entries are records, i.e. that X 1 < X 2 < ... < X N . Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.Records and sequences of records from random variables with a linear trend 2

We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff [1] and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution.We compute the record rate P n (c), defined as the probability for the nth value to be larger than all previous values, for a Gaussian jump distribution with standard deviation σ that is shifted by a constant drift c. For small drift, in the sense of c/σ ≪ n −1/2 , the correction to P n (c) grows proportional to arctan( √ n) and saturates at the value. For large n the record rate approaches a constant, which is approximately given by 1 − σ/ √ 2πc exp −c 2 /2σ 2 for c/σ ≫ 1. These asymptotic results carry over to other continuous jump distributions with finite variance. As an application, we compare our analytical results to the record statistics of 366 daily stock prices from the Standard & Poors 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.