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 $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq 1}$ is a sequence of independent random matrices taking values in $GL_d(\mathbb R)$, $d \geq 2$, with common distribution $\mu$. In this paper, under standard assumptions on $\mu$ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $\mu$ has a polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$ when $\mu$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n}$ when $\mu$ has a moment of order $4$, 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

CLT in Functional Linear Regression Models

International audienceWe propose in this work to derive a CLT in the functional linear regression model to get confidence sets for prediction based on functional linear regression. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made

Super optimal rates for nonparametric density estimation via projection estimators

In this paper, we study the problem of the nonparametric estimation of the marginal density f of a class of continuous time processes. To this aim, we use a projection estimator and deal with the integrated mean square risk. Under Castellana and Leadbetter's condition (Stoch. Proc. Appl. 21 (1986) 179), we show that our estimator reaches a parametric rate of convergence and coincides with the projection of the local time estimator. Discussions about the optimality of this condition are provided. We also deal with sampling schemes and the corresponding discretized processes.Castellana-Leadbetter's condition Continuous time projection estimator Markov processes Nonparametric estimation Local time Sampling

Deviation inequalities for dependent sequences with applications to strong approximations

In this paper, we give precise rates of convergence in the strong invariance principle for stationary sequences of bounded real-valued random variables satisfying weak dependence conditions. One of the main ingredients is a new Fuk-Nagaev type inequality for a class of weakly dependent sequences. We describe also several classes of processes to which our results apply

Deviation and concentration inequalities for dynamical systems with subexponential decay of correlation

We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples showing that the obtained upper bounds are essentially optimal

LARGE AND MODERATE DEVIATIONS FOR BOUNDED FUNCTIONS OF SLOWLY MIXING MARKOV CHAINS

International audienceWe consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities. These inequalities can be applied in various natural situations coming from probability theory or dynamical systems. Finally, we discuss examples from these various settings showing that our inequalities are sharp