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Christophe Cuny

Publications and source records attributed to Christophe Cuny.

At least 19 recordsLinked to original sources
arXiv2025

On the weak invariance principle for random fields with commuting filtrations under L1-projective criteria

We consider a field $f \circ T_1^{i_1} \circ \cdots \circ T_d^{i_d}$ where $T_1, \dots , T_d$ arecommuting transformations, one of them at least being ergodic. Considering the case of commuting filtrations, we are interested by giving sufficient ${\mathbb L}^1$-projective conditions ensuring that the normalized partial sums indexed by quadrants converge in distribution to a normal random variable. We also give sufficient conditions ensuring the weak invariance principle for the partial sums process. For the central limit theorem (CLT), the proof combines a truncated orthomartingale approximation with the CLT for orthomartingales due to Voln{\'y}. For the functional form, a new maximal inequality is needed and is obtained via truncation techniques, blocking arguments and orthomartingale approximations. The case of completely commuting transformations in the sense of Gordin can be handled in a similar way. Application to bounded Lipschitz functions of linear fields whose innovations have moments of a logarithmic order will be provided, as well as an application to completely commuting endomorphisms of the m-torus. In the latter case, the conditions can be expressed in terms of the ${\mathbb L}^1$-modulus of continuity of $f$.

math.PR↗
arXiv2023

On the growth rate of powers of a strongly Kreiss bounded operator on $L^p$-spaces

Let $T$ be a strongly Kreiss bounded linear operator on $L^p$. We obtain a bound on the rate of growth of the norms of the powers of $T$. The bound is optimal with respect to the polynomial scale. The proof makes use of Fourier multipliers, in particular of the Littlewood-Paley inequalities on arbitrary intervals as initiated by Rubio de Francia and developed by Kislyakov and Parilov.

math.FA↗
arXiv2023

Wiener's lemma along primes and other subsequences

Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $μ$ on the unit circle via the behavior of its Fourier coefficients $\hatμ(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results of Goldstein and Goldstein, Nagy. The results are complemented by some open questions and indication of interesting research directions. After this paper had been published, it was pointed out to us by Emmanuel Lesigne and Máté Wierdl that there is a gap in the Example on return times sequences along polynomials on page 13. Indeed, to make the argument there work, one needs a Wiener-Wintner type result for polynomial averages with a precise information about the limit, and this is presently out of reach.

math.FA↗
arXiv2020

Resolvent conditions and growth of powers of operators

Following Bermúdez et al. (ArXiv: 1706.03638v1), we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Cesàro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Cesàro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\|T^n\|=O(n/\sqrt {\log n})$; if $T$ is absolutely Cesàro bounded, $\|T^n\|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\|T^n\|=O((\log n)^κ)$ for some $κ>0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order $α$ converge strongly when $α>1$.

math.DS↗
arXiv2020

Resolvent conditions and growth of powers of operators on $L^p$ spaces

Let $T$ be a bounded linear operator on $L^p$. We study the rate of growth of the norms of the powers of $T$ under resolvent conditions or Cesàro boundedness assumptions. Actually the relevant properties of $L^p$ spaces in our study are their type and cotype, and for $1<p<\infty$, the fact that they are UMD. Some of the proofs make use of Fourier multipliers on Banach spaces, which explains why UMD spaces come into play.

math.FA↗
arXiv2020

On the convergence of bilateral ergodic averages

We study the almost sure convergence of bilateral ergodic averages for not necessarily integrable functions and relate it to the ones of the forward and backward averages, hence complementing results of Woś and the second named author. In the case of convergence, using results of Furstenberg on double recurrence, we prove oscillations of the bilateral ergodic averages around the limit.

math.DS↗
arXiv2019

Rates of convergence in invariance principles for random walks on linear groups via martingale methods

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of Wasserstein distance and in the almost sure invariance principle for R d-valued martingales with stationary increments having moments of order p $\in$]2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups.

math.PR↗
arXiv2017

An alternative to the coupling of Berkes-Liu-Wu for strong approximations

In this paper we propose an alternative to the coupling of Berkes, Liu and Wu [1] to obtain strong approximations for partial sums of dependent sequences. The main tool is a new Rosen-thal type inequality expressed in terms of the coupling coefficients. These coefficients are well suited to some classes of Markov chains or dynamical systems, but they also give new results for smooth functions of linear processes.

math.PR↗
arXiv2017

On Nörlund summation and Ergodic Theory, with applications to power series of Hilbert contractions

We show that if ${\bf a}=(a_n)_{n\in \N}$ is a good weight for the dominated weighted ergodic theorem in $L^p$, $p>1$, then the Nörlund matrix $N_{\bf a}=\{a_{i-j}/A_i\}_{0\le j\le i}$, $A_i=\sum_{k=0}^i |a_k|$ is bounded on $\ell^p(\N)$. We study the regularity (convergence in norm, almost everywhere) of operators in ergodic theory: power series of Hilbert contractions, and power series $\sum_{n\in \N} a_nP_nf $ of $L^2$-contractions, and establish similar tight relations with the Nörlund operator associated to the modulus coefficient sequence $(|a_n|)_{n\in \N}$.

math.FA↗
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

Pointwise convergence of almost periodic Fourier series and associated series of dilates

Let $\mathcal S^2$ be the Stepanov space and let $ λ_n\uparrow\infty$. Let $(a_n)_{n\ge 1}$ be satisfying Wiener's condition $A:= \sum_{n\ge 1} \big(\sum_{k\, :\, n\le λ_k \le n+1}|a_k|\big)^2 <\infty$. We prove that $\big\| \sup_{N\ge 1} \big|\sum_{n=1}^Na_n{\rm e}^{iλ_n t}\big| \, \big\|_{\mathcal S^2}\le C\, A^{1/2} $ where $C>0$ denotes a universal constant. Moreover, the series $\sum_{n\ge 1} a_n{\rm e}^{itλ_n }$ converges for $λ$-a.e. $t\in \mathbb R$. This contains as a special case Hedenmalm and Saksman result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let $1\le p,q\le 2$ be such that $1/p+1/q=3/2$. Then for any sequence $(α_n)_{n\ge 1}$ and $(β_n)_{n\ge 1}$ of complex numbers such that $K:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le λ_k< n+1}|α_k|\,\big)^p <\infty$ and $L:=\sum_{n\ge 1} \big(\sum_{k\,:\, n\le μ_k< n+1} |β_k|\,\big)^q <\infty$, we have $$ \Big\|\sup_{N\ge 1} \big|\sum_{n=1}^N α_n D(λ_n t)\big|\, \Big\|_{\mathcal S^2} \le C\, K^{1/p}\, L^{1/q } $$ where $D(t)= \sum_{n\ge 1}β_n {\rm e}^{iμ_n t}$ is defined in $\mathcal S^2$. Moreover, the series $\sum_{n\ge 1} α_n D(λ_nt)$ converges in $\mathcal S^2$ and for $λ$-a.e. $t\in \mathbb R$. We further show that if $\{λ_k, k\ge 1\}$ satisfies the following condition $$\sum_{ k\not=\ell\,,\, k'\not=\ell'\atop (k,\ell)\neq(k',\ell')}\big(1-|(λ_k-λ_\ell)-(λ_{k'}-λ_{\ell'}) |\big)_+^2 \, <\infty,$$ then the series $\sum_{k} a_k {\rm e}^{iλ_kt}$ converges on a set of positive Lebesgue measure, only if the series $\sum_{k=1}^\infty |a_k|^2$ converges. The above condition is in particular fulfilled when $\{λ_k, k\ge 1\}$ is a Sidon sequence.

math.CA↗
arXiv2016

Limit Theorems for the Left Random Walk on GLd (R)

Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of GLd (R), such as the Marcinkiewicz-Zygmund strong law of large numbers, the CLT (with rates in Wasserstein's distances) and almost sure invariance principles with rates.

math.PR↗
arXiv2016

Limit theorems under the Maxwell-Woodroofe condition in Banach spaces

We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers. Those results actually hold in the context of Banach valued stationary processes, including the case of $L^r$-valued random variables, with $1\le r<\infty$. In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \cite{PU}. Our results extend to non-adapted processes as well, and, partly to stationary processes arising from dynamical systems. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.

math.PR↗
arXiv2016

On the Ritt property and weak type maximal inequalities for convolution powers on $\ell^1(\Z)$

In this paper we study the behaviour of convolution powers of probability measures $μ$ on $\Z$, such that $(μ(n))_{n\in \N}$ is completely monotone or such that $ν$ is centered with a second moment. In particular we exhibit many new examples of probability measures on $\Z$ having the so called Ritt property and whose convolution powers satisfy weak type maximal inequalities in $\ell^1(\Z)$.

math.PR↗
arXiv2015

A compact LIL for martingales in $2$-smooth Banach spaces with applications

We prove the compact law of the iterated logarithm for stationary and ergodic differences of (reverse or not) martingales taking values in a separable $2$-smooth Banach space (for instance a Hilbert space). Then, in the martingale case, the almost sure invariance principle is derived from a result of Berger. From those results, we deduce the almost sure invariance principle for stationary processes under the Hannan condition and the compact law of the iterated logarithm for stationary processes arising from non-invertible dynamical systems. Those results for stationary processes are new, even in the real valued case. We also obtain the Marcinkiewicz-Zygmund strong law of large numbers for stationary processes with values in some smooth Banach spaces. Applications to several situations are given.

math.PR↗
arXiv2014

Ergodic theorems with arithmetical weights

We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},ν,τ)$ and any $f\in L^p(ν)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(τ^k x)$$ exists $ν$-almost everywhere. We also obtain similar results for other arithmetical functions, like $θ(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the circle method based on the shift model.

math.DS↗