68,946 research outputs found
Generalised extreme value statistics and sum of correlated variables
We show that generalised extreme value statistics -the statistics of the k-th
largest value among a large set of random variables- can be mapped onto a
problem of random sums. This allows us to identify classes of non-identical and
(generally) correlated random variables with a sum distributed according to one
of the three (k-dependent) asymptotic distributions of extreme value
statistics, namely the Gumbel, Frechet and Weibull distributions. These
classes, as well as the limit distributions, are naturally extended to real
values of k, thus providing a clear interpretation to the onset of Gumbel
distributions with non-integer index k in the statistics of global observables.
This is one of the very few known generalisations of the central limit theorem
to non-independent random variables. Finally, in the context of a simple
physical model, we relate the index k to the ratio of the correlation length to
the system size, which remains finite in strongly correlated systems.Comment: To appear in J.Phys.
Fast computation by block permanents of cumulative distribution functions of order statistics from several populations
The joint cumulative distribution function for order statistics arising from
several different populations is given in terms of the distribution function of
the populations. The computational cost of the formula in the case of two
populations is still exponential in the worst case, but it is a dramatic
improvement compared to the general formula by Bapat and Beg. In the case when
only the joint distribution function of a subset of the order statistics of
fixed size is needed, the complexity is polynomial, for the case of two
populations.Comment: 21 pages, 3 figure
Statistics of extremal intensities for Gaussian interfaces
The extremal Fourier intensities are studied for stationary
Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We
calculate the probability distribution of the maximal intensity and find that,
generically, it does not coincide with the distribution of the integrated power
spectrum (i.e. roughness of the surface), nor does it obey any of the known
extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit
distribution is, however, recovered in three cases: (i) in the non-dispersive
(white noise) limit, (ii) for high dimensions, and (iii) when only
short-wavelength modes are kept. In the last two cases the limit distribution
emerges in novel scenarios.Comment: 15 pages, including 7 ps figure
Extreme value distributions for weakly correlated fitnesses in block model
We study the limit distribution of the largest fitness for two models of
weakly correlated and identically distributed random fitnesses. The correlated
fitness is given by a linear combination of a fixed number of independent
random variables drawn from a common parent distribution. We find that for
certain class of parent distributions, the extreme value distribution for
correlated random variables can be related either to one of the known limit
laws for independent variables or the parent distribution itself. For other
cases, new limiting distributions appear. The conditions under which these
results hold are identified.Comment: Expanded, added reference
Continuous-time statistics and generalized relaxation equations
Using two simple examples, the continuous-time random walk as well as a two
state Markov chain, the relation between generalized anomalous relaxation
equations and semi-Markov processes is illustrated. This relation is then used
to discuss continuous-time random statistics in a general setting, for
statistics of convolution-type. Two examples are presented in some detail: the
sum statistic and the maximum statistic.Comment: 12 pages, submitted to EPJ
Order Statistics and Benford's Law
Fix a base B and let zeta have the standard exponential distribution; the
distribution of digits of zeta base B is known to be very close to Benford's
Law. If there exists a C such that the distribution of digits of C times the
elements of some set is the same as that of zeta, we say that set exhibits
shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1,
>..., X_N be independent identically distributed random variables. If the X_i's
are drawn from the uniform distribution on [0,L], then as N\to\infty the
distribution of the digits of the differences between adjacent order statistics
converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1).
By differentiating the cumulative distribution function of the logarithms
modulo 1, applying Poisson Summation and then integrating the resulting
expression, we derive rapidly converging explicit formulas measuring the
deviations from Benford's Law. Fix a delta in (0,1) and choose N independent
random variables from any compactly supported distribution with uniformly
bounded first and second derivatives and a second order Taylor series expansion
at each point. The distribution of digits of any N^\delta consecutive
differences \emph{and} all N-1 normalized differences of the order statistics
exhibit shifted exponential behavior. We derive conditions on the probability
density which determine whether or not the distribution of the digits of all
the un-normalized differences converges to Benford's Law, shifted exponential
behavior, or oscillates between the two, and show that the Pareto distribution
leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical
simulations on shifted exponential behavior have been suppressed (though are
available from the authors upon request). Version 4: a referee pointed out
that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has
now been adde
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