1,419 research outputs found
On some properties of contracting matrices
The concepts of paracontracting, pseudocontracting and nonexpanding operators
have been shown to be useful in proving convergence of asynchronous or parallel
iteration algorithms. The purpose of this paper is to give characterizations of
these operators when they are linear and finite-dimensional. First we show that
pseudocontractivity of stochastic matrices with respect to sup-norm is
equivalent to the scrambling property, a concept first introduced in the study
of inhomogeneous Markov chains. This unifies results obtained independently
using different approaches. Secondly, we generalize the concept of
pseudocontractivity to set-contractivity which is a useful generalization with
respect to the Euclidean norm. In particular, we demonstrate non-Hermitian
matrices that are set-contractive for ||.||_2, but not pseudocontractive for
||.||_2 or sup-norm. For constant row sum matrices we characterize
set-contractivity using matrix norms and matrix graphs. Furthermore, we prove
convergence results in compositions of set-contractive operators and illustrate
the differences between set-contractivity in different norms. Finally, we give
an application to the global synchronization in coupled map lattices.Comment: 17 page
Subgeometric ergodicity of strong Markov processes
We derive sufficient conditions for subgeometric f-ergodicity of strongly
Markovian processes. We first propose a criterion based on modulated moment of
some delayed return-time to a petite set. We then formulate a criterion for
polynomial f-ergodicity in terms of a drift condition on the generator.
Applications to specific processes are considered, including Langevin tempered
diffusions on R^n and storage models.Comment: Published at http://dx.doi.org/10.1214/105051605000000115 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Ergodicity and mixing bounds for the Fisher-Snedecor diffusion
We consider the Fisher-Snedecor diffusion; that is, the Kolmogorov-Pearson
diffusion with the Fisher-Snedecor invariant distribution. In the nonstationary
setting, we give explicit quantitative rates for the convergence rate of
respective finite-dimensional distributions to that of the stationary
Fisher-Snedecor diffusion, and for the -mixing coefficient of this
diffusion. As an application, we prove the law of large numbers and the central
limit theorem for additive functionals of the Fisher-Snedecor diffusion and
construct -consistent and asymptotically normal estimators for the
parameters of this diffusion given its nonstationary observation.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ453 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels
Let f be a classical holomorphic newform of level q and even weight k. We
show that the pushforward to the full level modular curve of the mass of f
equidistributes as qk -> infinity. This generalizes known results in the case
that q is squarefree. We obtain a power savings in the rate of equidistribution
as q becomes sufficiently "powerful" (far away from being squarefree), and in
particular in the "depth aspect" as q traverses the powers of a fixed prime.
We compare the difficulty of such equidistribution problems to that of
corresponding subconvexity problems by deriving explicit extensions of Watson's
formula to certain triple product integrals involving forms of non-squarefree
level. By a theorem of Ichino and a lemma of Michel--Venkatesh, this amounts to
a detailed study of Rankin--Selberg integrals int|f|^2 E attached to newforms f
of arbitrary level and Eisenstein series E of full level.
We find that the local factors of such integrals participate in many amusing
analogies with global L-functions. For instance, we observe that the mass
equidistribution conjecture with a power savings in the depth aspect is
equivalent to the union of a global subconvexity bound and what we call a
"local subconvexity bound"; a consequence of our local calculations is what we
call a "local Lindelof hypothesis".Comment: 43 pages; various minor corrections (many thanks to the referee) and
improvements in clarity and exposition. To appear in JAM
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