3,205 research outputs found
The Janson inequalities for general up-sets
Janson and Janson, Luczak and Rucinski proved several inequalities for the
lower tail of the distribution of the number of events that hold, when all the
events are up-sets (increasing events) of a special form - each event is the
intersection of some subset of a single set of independent events (i.e., a
principal up-set). We show that these inequalities in fact hold for arbitrary
up-sets, by modifying existing proofs to use only positive correlation,
avoiding the need to assume positive correlation conditioned on one of the
events.Comment: 5 pages. Added weighted varian
The lower tail: Poisson approximation revisited
The well-known "Janson's inequality" gives Poisson-like upper bounds for the
lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent
indicator random variables of a special form. We show that, for large
deviations, this inequality is optimal whenever X is approximately Poisson,
i.e., when the dependencies are weak. We also present correlation-based
approaches that, in certain symmetric applications, yield related conclusions
when X is no longer close to Poisson. As an illustration we, e.g., consider
subgraph counts in random graphs, and obtain new lower tail estimates,
extending earlier work (for the special case \eps=1) of Janson, Luczak and
Rucinski.Comment: 21 page
On covering by translates of a set
In this paper we study the minimal number of translates of an arbitrary
subset of a group needed to cover the group, and related notions of the
efficiency of such coverings. We focus mainly on finite subsets in discrete
groups, reviewing the classical results in this area, and generalizing them to
a much broader context. For example, we show that while the worst-case
efficiency when has elements is of order , for fixed and
large, almost every -subset of any given -element group covers
with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and
Algorithm
Upper tails for triangles
With the number of triangles in the usual (Erd\H{o}s-R\'enyi) random
graph , and , we show (for some )
\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].
This is tight up to the value of .Comment: 10 page
Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for
understanding the properties of suitable (convergent) sequences of
graphs in terms of a limiting object which may be represented by a symmetric
function on , i.e., a kernel or graphon. In this context it is
natural to wish to relate specific properties of the sequence to specific
properties of the kernel. Here we show that the kernel is monotone (i.e.,
increasing in both variables) if and only if the sequence satisfies a
`quasi-monotonicity' property defined by a certain functional tending to zero.
As a tool we prove an inequality relating the cut and norms of kernels of
the form with and monotone that may be of interest in its
own right; no such inequality holds for general kernels.Comment: 38 page
Central limit theorems for patterns in multiset permutations and set partitions
We use the recently developed method of weighted dependency graphs to prove
central limit theorems for the number of occurrences of any fixed pattern in
multiset permutations and in set partitions. This generalizes results for
patterns of size 2 in both settings, obtained by Canfield, Janson and
Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses
journal layou
Stein's method and stochastic analysis of Rademacher functionals
We compute explicit bounds in the Gaussian approximation of functionals of
infinite Rademacher sequences. Our tools involve Stein's method, as well as the
use of appropriate discrete Malliavin operators. Although our approach does not
require the classical use of exchangeable pairs, we employ a chaos expansion in
order to construct an explicit exchangeable pair vector for any random variable
which depends on a finite set of Rademacher variables. Among several examples,
which include random variables which depend on infinitely many Rademacher
variables, we provide three main applications: (i) to CLTs for multilinear
forms belonging to a fixed chaos, (ii) to the Gaussian approximation of
weighted infinite 2-runs, and (iii) to the computation of explicit bounds in
CLTs for multiple integrals over sparse sets. This last application provides an
alternate proof (and several refinements) of a recent result by Blei and
Janson.Comment: 35 pages + Appendix. New version: some inaccuracies in Sect. 6
correcte
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