51 research outputs found
Comparison between criteria leading to the weak invariance principle
The aim of this paper is to compare various criteria leading to the central
limit theorem and the weak invariance principle. These criteria are the
martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk
SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab.
Theory Related Fields 110 (1998), which was subsequently improved by Dedecker
and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the
condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later
improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in
every ergodic dynamical system with positive entropy, if we consider two of
these criteria, we can find a function in satisfying the first
but not the second.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP123 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
A counter example to central limit theorem in Hilbert spaces under a strong mixing condition
We show that in a separable infinite dimensional Hilbert space, uniform
integrability of the square of the norm of normalized partial sums of a
strictly stationary sequence, together with a strong mixing condition, does not
guarantee the central limit theorem.Comment: 12 pages, Electronic Communications in Probability, Volume 19, 201
New Techniques for Empirical Process of Dependent Data
We present a new technique for proving empirical process invariance principle
for stationary processes . The main novelty of our approach
lies in the fact that we only require the central limit theorem and a moment
bound for a restricted class of functions , not containing
the indicator functions. Our approach can be applied to Markov chains and
dynamical systems, using spectral properties of the transfer operator. Our
proof consists of a novel application of chaining techniques
Limit theorems for weighted Bernoulli random fields under Hannan's condition
Consider a Bernoulli random field satisfying the Hannan's condition.
Recently, invariance principles for partial sums of random fields over
rectangular index sets are established. In this note we complement previous
results by investigating limit theorems for weighted Bernoulli random fields,
including central limit theorems for partial sums over arbitrary index sets and
invariance principles for Gaussian random fields. Most results improve earlier
ones on Bernoulli random fields under Wu's condition, which is stronger than
Hannan's condition.Comment: 23 pages. Section 1 rewritten. Theorem 4.3 modified. Remark 4.4 adde
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