19 research outputs found
Renormalization flow in extreme value statistics
The renormalization group transformation for extreme value statistics of
independent, identically distributed variables, recently introduced to describe
finite size effects, is presented here in terms of a partial differential
equation (PDE). This yields a flow in function space and gives rise to the
known family of Fisher-Tippett limit distributions as fixed points, together
with the universal eigenfunctions around them. The PDE turns out to handle
correctly distributions even having discontinuities. Remarkably, the PDE admits
exact solutions in terms of eigenfunctions even farther from the fixed points.
In particular, such are unstable manifolds emanating from and returning to the
Gumbel fixed point, when the running eigenvalue and the perturbation strength
parameter obey a pair of coupled ordinary differential equations. Exact
renormalization trajectories corresponding to linear combinations of
eigenfunctions can also be given, and it is shown that such are all solutions
of the PDE. Explicit formulas for some invariant manifolds in the Fr\'echet and
Weibull cases are also presented. Finally, the similarity between
renormalization flows for extreme value statistics and the central limit
problem is stressed, whence follows the equivalence of the formulas for Weibull
distributions and the moment generating function of symmetric L\'evy stable
distributions.Comment: 21 pages, 9 figures. Several typos and an upload error corrected.
Accepted for publication in JSTA
Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution
We introduce an alternative definition of the relative height h^\kappa(x) of
a one-dimensional fluctuating interface indexed by a continuously varying real
paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to
the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the
spatially averaged height for \kappa = 1. We compute exactly the distribution
P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of
finite size L and periodic boundary conditions. One finds that it takes the
scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the
scaling function f^\kappa(x) interpolates between the Rayleigh distribution for
\kappa=0 and the Airy distribution for \kappa=1, the latter being the
probability distribution of the area under a Brownian excursion over the unit
interval. For arbitrary \kappa, one finds that it is related to, albeit
different from, the distribution of the area restricted to the interval [0,
\kappa] under a Brownian excursion over the unit interval.Comment: 25 pages, 4 figure
Renormalization group theory for finite-size scaling in extreme statistics
We present a renormalization group (RG) approach to explain universal
features of extreme statistics, applied here to independent, identically
distributed variables. The outlines of the theory have been described in a
previous Letter, the main result being that finite-size shape corrections to
the limit distribution can be obtained from a linearization of the RG
transformation near a fixed point, leading to the computation of stable
perturbations as eigenfunctions. Here we show details of the RG theory which
exhibit remarkable similarities to the RG known in statistical physics. Besides
the fixed points explaining universality, and the least stable eigendirections
accounting for convergence rates and shape corrections, the similarities
include marginally stable perturbations which turn out to be generic for the
Fisher-Tippett-Gumbel class. Distribution functions containing unstable
perturbations are also considered. We find that, after a transitory divergence,
they return to the universal fixed line at the same or at a different point
depending on the type of perturbation.Comment: 15 pages, 8 figures, to appear in Phys. Rev.
Distribution of the time at which N vicious walkers reach their maximal height
We study the extreme statistics of N non-intersecting Brownian motions
(vicious walkers) over a unit time interval in one dimension. Using
path-integral techniques we compute exactly the joint distribution of the
maximum M and of the time \tau_M at which this maximum is reached. We focus in
particular on non-intersecting Brownian bridges ("watermelons without wall")
and non-intersecting Brownian excursions ("watermelons with a wall"). We
discuss in detail the relationships between such vicious walkers models in
watermelons configurations and stochastic growth models in curved geometry on
the one hand and the directed polymer in a disordered medium (DPRM) with one
free end-point on the other hand. We also check our results using numerical
simulations of Dyson's Brownian motion and confront them with numerical
simulations of the Polynuclear Growth Model (PNG) and of a model of DPRM on a
discrete lattice. Some of the results presented here were announced in a recent
letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].Comment: 30 pages, 12 figure
Extreme value distributions and Renormalization Group
In the classical theorems of extreme value theory the limits of suitably
rescaled maxima of sequences of independent, identically distributed random
variables are studied. So far, only affine rescalings have been considered. We
show, however, that more general rescalings are natural and lead to new limit
distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The
problem is approached using the language of Renormalization Group
transformations in the space of probability densities. The limit distributions
are fixed points of the transformation and the study of the differential around
them allows a local analysis of the domains of attraction and the computation
of finite-size corrections.Comment: 16 pages, 5 figures. Final versio
Finite-size scaling in extreme statistics
We study the convergence and shape correction to the limit distributions of
extreme values due to the finite size (FS) of data sets. A renormalization
method is introduced for the case of independent, identically distributed (iid)
variables, showing that the iid universality classes are subdivided according
to the exponent of the FS convergence, which determines the leading order FS
shape correction function as well. We find that, for the correlated systems of
subcritical percolation and 1/f^alpha stationary (alpha<1) noise, the iid shape
correction compares favorably to simulations. Furthermore, for the strongly
correlated regime (alpha>1) of 1/f^alpha noise, the shape correction is
obtained in terms of the limit distribution itself.Comment: 4 pages, 3 figure
Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
We use the Real Space Renormalization Group (RSRG) method to study extreme
value statistics for a variety of Brownian motions, free or constrained such as
the Brownian bridge, excursion, meander and reflected bridge, recovering some
standard results, and extending others. We apply the same method to compute the
distribution of extrema of Bessel processes. We briefly show how the continuous
time random walk (CTRW) corresponds to a non standard fixed point of the RSRG
transformation.Comment: 24 pages, 5 figure
Universal Order Statistics of Random Walks
We study analytically the order statistics of a time series generated by the
successive positions of a symmetric random walk of n steps with step lengths of
finite variance \sigma^2. We show that the statistics of the gap
d_{k,n}=M_{k,n} -M_{k+1,n} between the k-th and the (k+1)-th maximum of the
time series becomes stationary, i.e, independent of n as n\to \infty and
exhibits a rich, universal behavior. The mean stationary gap (in units of
\sigma) exhibits a universal algebraic decay for large k,
/\sigma\sim 1/\sqrt{2\pi k}, independent of the details of the
jump distribution. Moreover, the probability density (pdf) of the stationary
gap exhibits scaling, Proba.(d_{k,\infty}=\delta)\simeq (\sqrt{k}/\sigma)
P(\delta \sqrt{k}/\sigma), in the scaling regime when \delta\sim
\simeq \sigma/\sqrt{2\pi k}. The scaling function P(x) is
universal and has an unexpected power law tail, P(x) \sim x^{-4} for large x.
For \delta \gg the scaling breaks down and the pdf gets cut-off
in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual
multi-scaling behavior.Comment: 5 pages, 3 figures. Revised version, typos corrected. Accepted for
publication in Physical Review Letter
Renormalization flow for extreme value statistics of random variables raised to a varying power
Using a renormalization approach, we study the asymptotic limit distribution
of the maximum value in a set of independent and identically distributed random
variables raised to a power q(n) that varies monotonically with the sample size
n. Under these conditions, a non-standard class of max-stable limit
distributions, which mirror the classical ones, emerges. Furthermore a
transition mechanism between the classical and the non-standard limit
distributions is brought to light. If q(n) grows slower than a characteristic
function q*(n), the standard limit distributions are recovered, while if q(n)
behaves asymptotically as k.q*(n), non-standard limit distributions emerge.Comment: 21 pages, 1 figure,final version, to appear in Journal of Physics
Systemic importance of financial institutions: regulations, research, open issues, proposals
In the field of risk management, scholars began to bring together the quantitative methodologies with the banking management issues about 30 years ago, with a special focus on market, credit and operational risks. After the systemic effects of banks defaults during the recent financial crisis,
and despite a huge amount of literature in the last years concerning the systemic risk, no standard methodologies have been set up to now. Even the new Basel 3 regulation has adopted a heuristic indicator-based approach, quite far from an effective quantitative tool. In this paper, we refer to the different pieces of the puzzle: definition of systemic risk, a set of coherent and useful measures, the computability of these measures, the data set structure. In this challenging field, we aim to build a comprehensive picture of the state of the art, to illustrate the open issues, and to outline some paths for a more successful future research. This work appropriately integrates other useful surveys and it is directed to both academic researchers and practitioners