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 Z2\mathbb Z^2

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 ${\mathbb Z}^2$. 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 ${\mathbb Z}[i]$, 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 ${\mathbb Z}^2$, 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