6,513 research outputs found
Modified empirical CLT's under only pre-Gaussian conditions
We show that a modified Empirical process converges to the limiting Gaussian
process whenever the limit is continuous. The modification depends on the
properties of the limit via Talagrand's characterization of the continuity of
Gaussian processes.Comment: Published at http://dx.doi.org/10.1214/074921706000000833 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Weak universality of dynamical : non-Gaussian noise
We consider a class of continuous phase coexistence models in three spatial
dimensions. The fluctuations are driven by symmetric stationary random fields
with sufficient integrability and mixing conditions, but not necessarily
Gaussian. We show that, in the weakly nonlinear regime, if the external
potential is a symmetric polynomial and a certain average of it exhibits
pitchfork bifurcation, then these models all rescale to near their
critical point.Comment: 37 pages; updated introduction and reference
Stochastic PDEs, Regularity Structures, and Interacting Particle Systems
These lecture notes grew out of a series of lectures given by the second
named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main
aim is to explain some aspects of the theory of "Regularity structures"
developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to
study well-posedness for a class of stochastic PDEs that could not be treated
previously. Prominent examples include the KPZ equation as well as the dynamic
model. Such equations can be expanded into formal perturbative
expansions. Roughly speaking the theory of regularity structures provides a way
to truncate this expansion after finitely many terms and to solve a fixed point
problem for the "remainder". The key ingredient is a new notion of "regularity"
which is based on the terms of this expansion.Comment: Fixed typo
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