379,668 research outputs found
On the functional central limit theorem via martingale approximation
In this paper, we develop necessary and sufficient conditions for the
validity of a martingale approximation for the partial sums of a stationary
process in terms of the maximum of consecutive errors. Such an approximation is
useful for transferring the conditional functional central limit theorem from
the martingale to the original process. The condition found is simple and well
adapted to a variety of examples, leading to a better understanding of the
structure of several stochastic processes and their asymptotic behaviors. The
approximation brings together many disparate examples in probability theory. It
is valid for classes of variables defined by familiar projection conditions
such as the Maxwell--Woodroofe condition, various classes of mixing processes,
including the large class of strongly mixing processes, and for additive
functionals of Markov chains with normal or symmetric Markov operators.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ276 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Approximate Bayesian Computation for a Class of Time Series Models
In the following article we consider approximate Bayesian computation (ABC)
for certain classes of time series models. In particular, we focus upon
scenarios where the likelihoods of the observations and parameter are
intractable, by which we mean that one cannot evaluate the likelihood even
up-to a positive unbiased estimate. This paper reviews and develops a class of
approximation procedures based upon the idea of ABC, but, specifically
maintains the probabilistic structure of the original statistical model. This
idea is useful, in that it can facilitate an analysis of the bias of the
approximation and the adaptation of established computational methods for
parameter inference. Several existing results in the literature are surveyed
and novel developments with regards to computation are given
On similarity classes of well-rounded sublattices of
A lattice is called well-rounded if its minimal vectors span the
corresponding Euclidean space. In this paper we study the similarity classes of
well-rounded sublattices of . We relate the set of all such
similarity classes to a subset of primitive Pythagorean triples, and prove that
it has structure of a noncommutative infinitely generated monoid. We discuss
the structure of a given similarity class, and define a zeta function
corresponding to each similarity class. We relate it to Dedekind zeta of
, and investigate the growth of some related Dirichlet series,
which reflect on the distribution of well-rounded lattices. Finally, we
construct a sequence of similarity classes of well-rounded sublattices of
, which gives good circle packing density and converges to the
hexagonal lattice as fast as possible with respect to a natural metric we
define.Comment: 27 pages, 2 figures; added a lemma on Diophantine approximation by
quotients of Pythagorean triples; final version to be published in Journal of
Number Theor
- …