47,709 research outputs found
Central limit theorem for Fourier transforms of stationary processes
We consider asymptotic behavior of Fourier transforms of stationary ergodic
sequences with finite second moments. We establish a central limit theorem
(CLT) for almost all frequencies and also an annealed CLT. The theorems hold
for all regular sequences. Our results shed new light on the foundation of
spectral analysis and on the asymptotic distribution of periodogram, and it
provides a nice blend of harmonic analysis, theory of stationary processes and
theory of martingales.Comment: Published in at http://dx.doi.org/10.1214/10-AOP530 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Geometry of logarithmic forms and deformations of complex structures
We present a new method to solve certain -equations for
logarithmic differential forms by using harmonic integral theory for currents
on Kahler manifolds. The result can be considered as a -lemma
for logarithmic forms. As applications, we generalize the result of Deligne
about closedness of logarithmic forms, give geometric and simpler proofs of
Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral
sequences at -level, as well as certain injectivity theorem on compact
Kahler manifolds.
Furthermore, for a family of logarithmic deformations of complex structures
on Kahler manifolds, we construct the extension for any logarithmic
-form on the central fiber and thus deduce the local stability of log
Calabi-Yau structure by extending an iteration method to the logarithmic forms.
Finally we prove the unobstructedness of the deformations of a log Calabi-Yau
pair and a pair on a Calabi-Yau manifold by differential geometric method.Comment: Several typos have been fixed. Final version to appear in Journal of
Algebraic Geometr
On Transformations of Markov Chains and Poisson Boundary
A discrete-time Markov chain can be transformed into a new Markov chain by
looking at its states along iterations of an almost surely finite stopping
time. By the optional stopping theorem, any bounded harmonic function with
respect to the transition function of the original chain is harmonic with
respect to the transition function of the transformed chain. The reverse
inclusion is in general not true. Our main result provides a sufficient
condition on the stopping time which guarantees that the space of bounded
harmonic functions for the transformed chain embeds in the space of bounded
harmonic sequences for the original chain. We also obtain a similar result on
positive unbounded harmonic functions, under some additional conditions. Our
work was motivated by and is analogous to Forghani-Kaimanovich, the
well-studied case when the Markov chain is a random walk on a discrete group
Interpolation and harmonic majorants in big Hardy-Orlicz spaces
Free interpolation in Hardy spaces is caracterized by the well-known Carleson
condition. The result extends to Hardy-Orlicz spaces contained in the scale of
classical Hardy spaces , . For the Smirnov and the Nevanlinna
classes, interpolating sequences have been characterized in a recent paper in
terms of the existence of harmonic majorants (quasi-bounded in the case of the
Smirnov class). Since the Smirnov class can be regarded as the union over all
Hardy-Orlicz spaces associated with a so-called strongly convex function, it is
natural to ask how the condition changes from the Carleson condition in
classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of
this paper is to narrow down this gap from the Smirnov class to ``big''
Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences
for a class of Hardy-Orlicz spaces that carry an algebraic structure and that
are strictly bigger than . It turns out that the
interpolating sequences are again characterized by the existence of
quasi-bounded majorants, but now the weights of the majorants have to be in
suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz
spaces will also be discussed in the general situation. We finish the paper
with an example of a separated Blaschke sequence that is interpolating for
certain Hardy-Orlicz spaces without being interpolating for slightly smaller
ones.Comment: 19 pages, 2 figure
Orders of accumulation of entropy
For a continuous map of a compact metrizable space with finite
topological entropy, the order of accumulation of entropy of is a countable
ordinal that arises in the context of entropy structure and symbolic
extensions. We show that every countable ordinal is realized as the order of
accumulation of some dynamical system. Our proof relies on functional analysis
of metrizable Choquet simplices and a realization theorem of Downarowicz and
Serafin. Further, if is a metrizable Choquet simplex, we bound the ordinals
that appear as the order of accumulation of entropy of a dynamical system whose
simplex of invariant measures is affinely homeomorphic to . These bounds are
given in terms of the Cantor-Bendixson rank of \overline{\ex(M)}, the closure
of the extreme points of , and the relative Cantor-Bendixson rank of
\overline{\ex(M)} with respect to \ex(M). We also address the optimality of
these bounds.Comment: 48 page
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