30 research outputs found
A multivariate CLT for bounded decomposable random vectors with the best known rate
We prove a multivariate central limit theorem with explicit error bound on a
non-smooth function distance for sums of bounded decomposable -dimensional
random vectors. The decomposition structure is similar to that of Barbour,
Karo\'nski and Ruci\'nski (1989) and is more general than the local dependence
structure considered in Chen and Shao (2004). The error bound is of the order
, where is the dimension and is the
number of summands. The dependence on , namely , is the
best known dependence even for sums of independent and identically distributed
random vectors, and the dependence on , namely , is
optimal. We apply our main result to a random graph example.Comment: 12 page
A Berry-Esseen bound with applications to vertex degree counts in the Erd\H{o}s-R\'{e}nyi random graph
Applying Stein's method, an inductive technique and size bias coupling yields
a Berry-Esseen theorem for normal approximation without the usual restriction
that the coupling be bounded. The theorem is applied to counting the number of
vertices in the Erdos-Renyi random graph of a given degree.Comment: Published in at http://dx.doi.org/10.1214/12-AAP848 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
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