arXiv subjects

Florence Merlevède

Publications and source records attributed to Florence Merlevède.

At least 37 records · Page 2Linked to original sources

arXiv2017

On the Komlós, Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see [15] and [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in L p (p > 2) by a Wiener process with an error term of order o(n 1/p). Very recently, Berkes, Liu and Wu [3] extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in L p is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in L $\infty$ or in L 1), under which the strong approximation result holds with rate o(n 1/p). As we shall see our conditions are well adapted to a large variety of models, including left random walks on GL d (R), contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our L 1-coupling condition is in some sense optimal.

math.PR↗

arXiv2016

Large and moderate deviations for the left random walk on GL d (R)

Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements of GL d (R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of order p \textgreater{} 1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.

math.PR↗

arXiv2016

Density estimation for $β$-dependent sequences

We study the Lp-integrated risk of some classical estimators of the density, when the observations are drawn from a strictly stationary sequence. The results apply to a large class of sequences, which can be non-mixing in the sense of Rosenblatt and long-range dependent. The main probabilistic tool is a new Rosenthal-type inequality for partial sums of BV functions of the variables. As an application, we give the rates of convergence of regular Histograms, when estimating the invariant density of a class of expanding maps of the unit interval with a neutral fixed point at zero. These Histograms are plotted in the section devoted to the simulations.

math.ST↗

arXiv2016

A deviation bound for $α$-dependent sequences with applications to intermittent maps

We prove a deviation bound for the maximum of partial sums of functions of $α$-dependent sequences as defined in Dedecker, Gou{ë}zel and Merlev{è}de (2010). As a consequence, we extend the Rosenthal inequality of Rio (2000) for $α$-mixing sequences in the sense of Rosenblatt (1956) to the larger class of $α$-dependent sequences. Starting from the deviation inequality, we obtain upper bounds for large deviations and an H{ö}lderian invariance principle for the Donsker line. We illustrate our results through the example of intermittent maps of the interval, which are not $α$-mixing in the sense of Rosenblatt.

math.PR↗

arXiv2015

Bernstein type inequality for a class of dependent random matrices

In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. (2009) in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

math.PR↗

arXiv2015

Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $α$-dependent sequences

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $α$-dependent sequences. We prove some moments inequalities of order p for any p $\ge$ 1, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well known von Bahr-Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identically distributed random variables.

math.PR↗

arXiv2014

Moment bounds for dependent sequences in smooth Banach spaces

We prove a Marcinkiewicz-Zygmund type inequality for random variables taking values in a smooth Banach space. Next, we obtain some sharp concentration inequalities for the empirical measure of $\{T, T^2, \cdots, T^n\}$, on a class of smooth functions, when $T$ belongs to a class of nonuniformly expanding maps of the unit interval.

math.PR↗

arXiv2014

On martingale approximations and the quenched weak invariance principle

In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in ${\mathcal{H}}$ (a real and separable Hilbert space) admits an approximation, in ${\mathbb{L}}^p({\mathcal{H}})$, $p>1$, by a martingale with stationary differences, and we then estimate the error of approximation in ${\mathbb{L}}^p({\mathcal{H}})$. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz-Zygmund theorem, the moderate deviations principle and the rates in the central limit theorem in terms of Wasserstein distances. The conditions are well suited for a large variety of examples, including linear processes or various kinds of weak dependent or mixing processes. In addition, our approach suits well to investigate the quenched central limit theorem and its invariance principle via martingale approximation, and allows us to show that they hold under the so-called Maxwell-Woodroofe condition that is known to be optimal.

math.PR↗

arXiv2014

On the universality of spectral limit for random matrices with martingale differences entries

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko-Pastur law and other convergence results. In the stationary case we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko-Pastur one. Our results are well adapted to study several examples including non linear ARCH infinite random fields.

math.PR↗

arXiv2013

Strong approximation results for the empirical process of stationary sequences

We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than $n^{-1-δ}$ for some positive $δ$. This shows that our conditions are in some sense optimal.

math.PR↗

arXiv2013

Reflexive Operator Algebras on Banach Spaces

In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant.

math.FA↗

arXiv2013

Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples

The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio [J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Various applications are also provided.

math.PR↗

arXiv2013

A quenched weak invariance principle

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and non irreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, a theory which has interest in itself.

math.PR↗

arXiv2012

Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary part of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and technics from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.

math.PR↗

arXiv2012

Rates in the strong invariance principle for ergodic automorphisms of the torus

Let T be an ergodic automorphism of the d-dimensional torus. In the spirit of Le Borgne, we give conditions on the Fourier coeffi cients of a real valued function f under which the Birkhoff sums satis fy a strong invariance principle. Next, reinforcing the condition on the Fourier coeffi cients in a natural way, we obtain explicit rates of convergence in the strong invariance principle.

math.DS↗

arXiv2012

Rates of convergence in the strong invariance principle under projective criteria

We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety of examples, including mixing processes of different kinds. We present some applications to symmetric random walks on the circle, to functions of dependent sequences, and to a reversible Markov chain.

math.PR↗

arXiv2012

Bernstein inequality and moderate deviations under strong mixing conditions

In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coeficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.

math.PR↗

arXiv2011

Almost Sure Invariance Principles via Martingale Approximation

In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.

math.PR↗