3,320 research outputs found
Fully oscillating sequences and weighted multiple ergodic limit
We prove that fully oscillating sequences are orthogonal to multiple ergodic
realizations of affine maps of zero entropy on compact abelian groups. It is
more than what Sarnak's conjecture requires for these dynamical systems
A new metric between distributions of point processes
Most metrics between finite point measures currently used in the literature
have the flaw that they do not treat differing total masses in an adequate
manner for applications. This paper introduces a new metric that
combines positional differences of points under a closest match with the
relative difference in total mass in a way that fixes this flaw. A
comprehensive collection of theoretical results about and its
induced Wasserstein metric for point process distributions are
given, including examples of useful -Lipschitz continuous functions,
upper bounds for Poisson process approximation, and
upper and lower bounds between distributions of point processes of i.i.d.
points. Furthermore, we present a statistical test for multiple point pattern
data that demonstrates the potential of in applications.Comment: 20 pages, 2 figure
Zero biasing and a discrete central limit theorem
We introduce a new family of distributions to approximate for and a sum of independent
integer-valued random variables , , with finite
second moments, where, with large probability, is not concentrated on a
lattice of span greater than 1. The well-known Berry--Esseen theorem states
that, for a normal random variable with mean and variance
, provides a good approximation
to for of the form . However, for more
general , such as the set of all even numbers, the normal approximation
becomes unsatisfactory and it is desirable to have an appropriate discrete,
nonnormal distribution which approximates in total variation, and a
discrete version of the Berry--Esseen theorem to bound the error. In this
paper, using the concept of zero biasing for discrete random variables (cf.
Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a
new family of discrete distributions and provide a discrete version of the
Berry--Esseen theorem showing how members of the family approximate the
distribution of a sum of integer-valued variables in total variation.Comment: Published at http://dx.doi.org/10.1214/009117906000000250 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On approximation of Markov binomial distributions
For a Markov chain with the state space
, the random variable is said to follow a Markov
binomial distribution. The exact distribution of , denoted ,
is very computationally intensive for large (see Gabriel [Biometrika 46
(1959) 454--460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677--680])
and this paper concerns suitable approximate distributions for
when is stationary. We conclude that the negative binomial and
binomial distributions are appropriate approximations for when
is greater than and less than ,
respectively. Also, due to the unique structure of the distribution, we are
able to derive explicit error estimates for these approximations.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ194 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
The probability density quantile (pdQ) carries essential information
regarding shape and tail behavior of a location-scale family. Convergence of
repeated applications of the pdQ mapping to the uniform distribution is
investigated and new fixed point theorems are established. The Kullback-Leibler
divergences from uniformity of these pdQs are mapped and found to be
ingredients in power functions of optimal tests for uniformity against
alternative shapes.Comment: 13 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1605.0018
- …