8 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
A decade of weather extremes
The ostensibly large number of recent extreme weather events has triggered intensive discussions, both in- and outside the scientific community, on whether they are related to global warming. Here, we review the evidence and argue that for some types of extreme - notably heatwaves, but also precipitation extremes - there is now strong evidence linking specific events or an increase in their numbers to the human influence on climate. For other types of extreme, such as storms, the available evidence is less conclusive, but based on observed trends and basic physical concepts it is nevertheless plausible to expect an increase