2,135 research outputs found
Moderate deviations via cumulants
The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and -statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and random point fields.Comment: 24 page
Weighted dependency graphs
The theory of dependency graphs is a powerful toolbox to prove asymptotic
normality of sums of random variables. In this article, we introduce a more
general notion of weighted dependency graphs and give normality criteria in
this context. We also provide generic tools to prove that some weighted graph
is a weighted dependency graph for a given family of random variables.
To illustrate the power of the theory, we give applications to the following
objects: uniform random pair partitions, the random graph model ,
uniform random permutations, the symmetric simple exclusion process and
multilinear statistics on Markov chains. The application to random permutations
gives a bivariate extension of a functional central limit theorem of Janson and
Barbour. On Markov chains, we answer positively an open question of Bourdon and
Vall\'ee on the asymptotic normality of subword counts in random texts
generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces
How unproportional must a graph be?
Let be the maximum over all -vertex graphs of by how much
the number of induced copies of in differs from its expectation in the
binomial random graph with the same number of vertices as and with edge
probability . This may be viewed as a measure of how close is to being
-quasirandom. For a positive integer and , let be the
distance from to the nearest integer. Our main result is that,
for fixed and for large, the minimum of over -vertex
graphs has order of magnitude
provided that
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
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