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Christophe Cuny

Publications and source records attributed to Christophe Cuny.

26 records · Page 2Linked to original sources
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↗
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↗
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

Strong invariance principles with rate for "reverse" martingales and applications

In this paper, we obtain almost sure invariance principles with rate of order $n^{1/p}\log^βn$, $2< p\le 4$, for sums associated to a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance we treat several classes of uniformly expanding maps of the interval (for possibly unbounded functions). A general result for $ϕ$-dependent sequences is obtained in the course.

math.PR↗
arXiv2012

A quenched invariance principle for stationary processes

In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain.

math.PR↗
arXiv2009

Pointwise ergodic theorems with rate and application to limit theorems for stationary processes

We obtain pointwise ergodic theorems with rate under conditions expressed in terms of the convergence of series involving $\|\sum_{k=1} ^nf\circ θ^k\|_2$, improving previous results. Then, using known results on martingale approximation, we obtain some LIL for stationary ergodic processes and quenched central limit theorems for functional of Markov chains. The proofs are based on the use of the spectral theorem and, on a recent work of Zhao-Woodroofe extending a method of Derriennic-Lin.

math.PR↗
arXiv2006

On random almost periodic trigonometric polynomials and applications to ergodic theory

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(λ_k^{(1)}t_1+...+λ_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{λ_n^{(i)}:1\leq i\leq s\}$ are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered $\{X_n\}$ or bounded $\{X_n\}$ satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443--457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193--216] in several directions. As applications we derive conditions for uniform convergence of these sums on compact sets. We also obtain random ergodic theorems for finitely many commuting measure-preserving point transformations of a probability space. Finally, we show how some of our results allow to derive the Wiener--Wintner property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003) 1637--1654]) for certain functions on certain dynamical systems.

math.PR↗