14 research outputs found
Records and sequences of records from random variables with a linear trend
We consider records and sequences of records drawn from discrete time series
of the form , where the are independent and identically
distributed random variables and is a constant drift. For very small and
very large drift velocities, we investigate the asymptotic behavior of the
probability of a record occurring in the th step and the
probability that all entries are records, i.e. that . Our work is motivated by the analysis of temperature time series in
climatology, and by the study of mutational pathways in evolutionary biology.Comment: 21 pages, 7 figure
Correlations of record events as a test for heavy-tailed distributions
A record is an entry in a time series that is larger or smaller than all
previous entries. If the time series consists of independent, identically
distributed random variables with a superimposed linear trend, record events
are positively (negatively) correlated when the tail of the distribution is
heavier (lighter) than exponential. Here we use these correlations to detect
heavy-tailed behavior in small sets of independent random variables. The method
consists of converting random subsets of the data into time series with a
tunable linear drift and computing the resulting record correlations.Comment: Revised version, to appear in Physical Review Letter
Record statistics and persistence for a random walk with a drift
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.Comment: 51 pages, 22 figures. Published version (typos have been corrected
Record statistics for biased random walks, with an application to financial data
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 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 , defined as the probability for the th
value to be larger than all previous values, for a Gaussian jump distribution
with standard deviation that is shifted by a constant drift . For
small drift, in the sense of , the correction to
grows proportional to arctan and saturates at the value
. For large the record rate approaches a
constant, which is approximately given by
for .
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.Comment: 16 pages, 7 figure
Record-breaking temperatures reveal a warming climate
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. Copyright (C) EPLA, 201
Rounding Effects in Record Statistics
We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding to integer multiples of a discretization scale Delta > 0. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case, the record sequence becomes highly regular at long times