205 research outputs found

    Empirical central limit theorems for ergodic automorphisms of the torus

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    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

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    Let An=Δn⋯Δ1A_n= \varepsilon_n \cdots \varepsilon_1, where (Δn)n≄1(\varepsilon_n)_{n \geq 1} is a sequence of independent random matrices taking values in GLd(R) GL_d(\mathbb R), d≄2d \geq 2, with common distribution ÎŒ\mu. In this paper, under standard assumptions on ÎŒ\mu (strong irreducibility and proximality), we prove Berry-Esseen type theorems for log⁥(∄An∄)\log ( \Vert A_n \Vert) when ÎŒ\mu has a polynomial moment. More precisely, we get the rate ((log⁥n)/n)q/2−1((\log n) / n)^{q/2-1} when ÎŒ\mu has a moment of order q∈]2,3]q \in ]2,3] and the rate 1/n1/ \sqrt{n} when ÎŒ\mu has a moment of order 44, which significantly improves earlier results in this setting

    Rates in almost sure invariance principle for nonuniformly hyperbolic maps

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    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

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    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 mm. 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

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    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

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    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

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    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

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    We propose an algorithm to estimate the common density ss of a stationary process X1,...,XnX_1,...,X_n. We suppose that the process is either ÎČ\beta or τ\tau-mixing. We provide a model selection procedure based on a generalization of Mallows' CpC_p 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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