204 research outputs found
Empirical central limit theorems for ergodic automorphisms of the torus
Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a
continuous function from T^d to R^l. On the probability space T^d equipped with
the Lebesgue-Haar measure, we prove the weak convergence of the sequential
empirical process of the sequence (f o T^i)_{i \geq 1} under some mild
conditions on the modulus of continuity of f. The proofs are based on new limit
theorems and new inequalities for non-adapted sequences, and on new estimates
of the conditional expectations of f with respect to a natural filtration.Comment: 32 page
Berry-Esseen type bounds for the Left Random Walk on GL d (R) under polynomial moment conditions
Let , where is a sequence of independent random matrices taking values in , , with common distribution . In this paper,
under standard assumptions on (strong irreducibility and proximality), we
prove Berry-Esseen type theorems for when has a
polynomial moment. More precisely, we get the rate
when has a moment of order and the rate when
has a moment of order , which significantly improves earlier results
in this setting
Rates in almost sure invariance principle for nonuniformly hyperbolic maps
We prove the Almost Sure Invariance Principle (ASIP) with close to optimal
error rates for nonuniformly hyperbolic maps. We do not assume exponential
contraction along stable leaves, therefore our result covers in particular
slowly mixing invertible dynamical systems as Bunimovich flowers, billiards
with flat points as in Chernov and Zhang (2005) and Wojtkowski' (1990) system
of two falling balls. For these examples, the ASIP is a new result, not covered
by prior works for various reasons, notably because in absence of exponential
contraction along stable leaves, it is challenging to employ the so-called
Sinai's trick (Sinai 1972, Bowen 1975) of reducing a nonuniformly hyperbolic
system to a nonuniformly expanding one. Our strategy follows our previous
papers on the ASIP for nonuniformly expanding maps, where we build a
semiconjugacy to a specific renewal Markov shift and adapt the argument of
Berkes, Liu and Wu (2014). The main difference is that now the Markov shift is
two-sided, the observables depend on the full trajectory, both the future and
the past
Homogeneous variational problems: a minicourse
A Finsler geometry may be understood as a homogeneous variational problem,
where the Finsler function is the Lagrangian. The extremals in Finsler geometry
are curves, but in more general variational problems we might consider extremal
submanifolds of dimension . In this minicourse we discuss these problems
from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse
given at the sixth Bilateral Workshop on Differential Geometry and its
Applications, held in Ostrava in May 201
On symmetries of Chern-Simons and BF topological theories
We describe constructing solutions of the field equations of Chern-Simons and
topological BF theories in terms of deformation theory of locally constant
(flat) bundles. Maps of flat connections into one another (dressing
transformations) are considered. A method of calculating (nonlocal) dressing
symmetries in Chern-Simons and topological BF theories is formulated
Context Tree Selection: A Unifying View
The present paper investigates non-asymptotic properties of two popular
procedures of context tree (or Variable Length Markov Chains) estimation:
Rissanen's algorithm Context and the Penalized Maximum Likelihood criterion.
First showing how they are related, we prove finite horizon bounds for the
probability of over- and under-estimation. Concerning overestimation, no
boundedness or loss-of-memory conditions are required: the proof relies on new
deviation inequalities for empirical probabilities of independent interest. The
underestimation properties rely on loss-of-memory and separation conditions of
the process.
These results improve and generalize the bounds obtained previously. Context
tree models have been introduced by Rissanen as a parsimonious generalization
of Markov models. Since then, they have been widely used in applied probability
and statistics
Symmetries of Helmholtz forms and globally variational dynamical forms
Invariance properties of classes in the variational sequence suggested to
Krupka et al. the idea that there should exist a close correspondence between
the notions of variationality of a differential form and invariance of its
exterior derivative. It was shown by them that the invariance of a closed
Helmholtz form of a dynamical form is equivalent with local variationality of
the Lie derivative of the dynamical form, so that the latter is locally the
Euler--Lagrange form of a Lagrangian. We show that the corresponding local
system of Euler--Lagrange forms is variationally equivalent to a global
Euler--Lagrange form.Comment: Presented at QTS7 - Quantum Theory and Symmetries VII, Prague
7-13/08/201
Adaptive density estimation for stationary processes
We propose an algorithm to estimate the common density of a stationary
process . We suppose that the process is either or
-mixing. We provide a model selection procedure based on a generalization
of Mallows' and we prove oracle inequalities for the selected estimator
under a few prior assumptions on the collection of models and on the mixing
coefficients. We prove that our estimator is adaptive over a class of Besov
spaces, namely, we prove that it achieves the same rates of convergence as in
the i.i.d framework
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