In this paper, we obtain almost sure invariance principles with rate of order n 1/p log β n, 2 < p ≤ 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. (2010): 37E05, 37C30, 60F15. Mathematics Subject Classification

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 GL d (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.

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.Comment: Published in at http://dx.doi.org/10.1214/13-AOP856 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We obtain pointwise ergodic theorems with rate under conditions expressed in terms of the convergence of series involving n k=1 f •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.1991 Mathematics Subject Classification. Primary: 60F15 ; Secondary: 60F05.

Abstract. 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, A, ν, τ ) and any f ∈ L p (ν), p > 1, the limitexists 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 Js(n), s > 0. We use Bourgain's method, namely the circle method based on the shift model.