904 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
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
Approximating class approach for empirical processes of dependent sequences indexed by functions
We study weak convergence of empirical processes of dependent data
, indexed by classes of functions. Our results are especially
suitable for data arising from dynamical systems and Markov chains, where the
central limit theorem for partial sums of observables is commonly derived via
the spectral gap technique. We are specifically interested in situations where
the index class is different from the class of functions
for which we have good properties of the observables . We
introduce a new bracketing number to measure the size of the index class
which fits this setting. Our results apply to the empirical
process of data satisfying a multiple mixing condition. This
includes dynamical systems and Markov chains, if the Perron-Frobenius operator
or the Markov operator has a spectral gap, but also extends beyond this class,
for example, to ergodic torus automorphisms.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ525 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Interaction on Hypergraphs
Interaction on hypergraphs generalizes interaction on graphs, also known as pairwise local interaction. For games played on a hypergraph which are supermodular potential games, logit-perturbed best-response dynamics are studied. We find that the associated stochastically stable states form a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest stochastically stable state. In the special case of networking games, we obtain comparative statics results with respect to investment costs, for Nash equilibria of supermodular games as well as for Nash equilibria of submodular games.
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