94 research outputs found
A fourth moment inequality for functionals of stationary processes
In this paper, a fourth moment bound for partial sums of functional of
strongly ergodic Markov chain is established. This type of inequality plays an
important role in the study of empirical process invariance principle. This one
is specially adapted to the technique of Dehling, Durieu and Voln\'y (2008).
The same moment bound can be proved for dynamical system whose transfer
operator has some spectral properties. Examples of applications are given
Empirical Processes of Multidimensional Systems with Multiple Mixing Properties
We establish a multivariate empirical process central limit theorem for
stationary -valued stochastic processes under very weak
conditions concerning the dependence structure of the process. As an
application we can prove the empirical process CLT for ergodic torus
automorphisms. Our results also apply to Markov chains and dynamical systems
having a spectral gap on some Banach space of functions. Our proof uses a
multivariate extension of the techniques introduced by Dehling, Durieu and
Voln\'y \cite{DehDurVol09} in the univariate case. As an important technical
ingredient, we prove a th moment bound for partial sums in multiple
mixing systems.Comment: to be published in Stochastic Processes and their Application
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
From infinite urn schemes to decompositions of self-similar Gaussian processes
We investigate a special case of infinite urn schemes first considered by
Karlin (1967), especially its occupancy and odd-occupancy processes. We first
propose a natural randomization of these two processes and their
decompositions. We then establish functional central limit theorems, showing
that each randomized process and its components converge jointly to a
decomposition of certain self-similar Gaussian process. In particular, the
randomized occupancy process and its components converge jointly to the
decomposition of a time-changed Brownian motion , and the randomized odd-occupancy process and its components
converge jointly to a decomposition of fractional Brownian motion with Hurst
index . The decomposition in the latter case is a special case of
the decompositions of bi-fractional Brownian motions recently investigated by
Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed
as correlated random walks, and in particular as a complement to the model
recently introduced by Hammond and Sheffield (2013) as discrete analogues of
fractional Brownian motions.Comment: 25 page
Independence of Four Projective Criteria for the Weak Invariance Principle
Let be a regular stationary process for a given filtration.
The weak invariance principle holds under the condition
(see Hannan (1979)}, Dedecker and
Merlev\`ede (2003), Deddecker, Merlev\'ede and Voln\'y (2007)). In this paper,
we show that this criterion is independent of other known criteria: the
martingale-coboundary decomposition of Gordin (see Gordin (1969, 1973)), the
criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the condition
of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade and Utev
(2005), Voln\'y (2006, 2007)).Comment: 6 page
Empirical processes of iterated maps that contract on average
We consider a Markov chain obtained by random iterations of Lipschitz maps
chosen with a probability depending on the current position .
We assume this system has a property of "contraction on average", that is
for some . In the present
note, we study the weak convergence of the empirical process associated to this
Markov chain
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
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