Emmanuel Lesigne

Large deviations for martingales

Let (Xi) be a martingale difference sequence and Sn=[summation operator]i=1n Xi. We prove that if supi E(eXi) 0 such that [mu](Sn>n)[less-than-or-equals, slant]e-cn1/3; this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (Xi) is bounded in Lp, 2[less-than-or-equals, slant]p n)[less-than-or-equals, slant]cn-p/2 which is again optimal for strictly stationary and ergodic sequences of martingale differences. These estimations can be extended to martingale difference fields. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, [mu](Sn>n)=o(n1-p) for Xi[set membership, variant]Lp, 1[less-than-or-equals, slant]p 0.

Ergodicity of Rokhlin cocycles

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions.We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations.Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems.

Théorèmes ergodiques pour une translation sur un nilvariété

Let T be a translation defined on a nil-manifold N /Γ, where N is a nilpotent group of order two.

Almost sure central limit theorem for strictly stationary processes

On any aperiodic measure preserving system, there exists a square integrable function such that the associated stationary process satifies the Almost Sure Central Limit Theorem. Introduction The Almost Sure Central Limit Theorem (ASCLT), first formulated by Lévy in [9], has been studied by various authors at the end of the eighties ([6], [3], [10], [8]). This theorem gives conditions under which, for a sequence of random variables satisfying the Central Limit Theorem (CLT), the Gaussian asymptotic behaviour can be observed along individual trajectory of the process. In the Lacey and Philipp note [8], the ASCLT is stated under optimal hypotheses, and the proof is short and clear. Here is their result. (If x is a real number, notation δ(x) will be used for the Dirac mass at point x.) Theorem. Let (Xn)n≥1 be an independent and identically distributed sequence of square integrable real random variables with E(Xn) = 0 and E(X n) = 1. Almost surely, the sequence of probability distributions ( 1 logn n ∑ k=1 1 k δ ( X1 +X2 + · · ·+Xk √ k )) n≥1 converges weakly to the Gaussian law N(0, 1). Several authors, including Berkes and Dehling ([2]), Atlagh and Weber ([1]), and Lacey have observed that for i.i.d. sequences, the finite second moment condition is necessary for the ASCLT. So, in this context, necessary and sufficient conditions for the CLT and the ASCLT are the same. This paper is a contribution to the study of the general case of strictly stationary sequences. The question of the CLT for strictly stationary processes has been extensively studied, from various points of view. Given a probability measure preserving dynamical system (Ω, T , μ, T ) and a real measurable function f on Ω we say that Received by the editors June 14, 1998 and, in revised form, July 22, 1998. 1991 Mathematics Subject Classification. Primary 28D05, 60G10, 60F05.

Weak disjointness of measure-preserving dynamical systems

Two measure preserving dynamical systems are weakly disjoint if some pointwise convergence property is satisfied by ergodic averages on their direct product (a precise definition is given below). Disjointness implies weak disjointness. We start studying this new concept, both by stating some general properties and by giving various examples. The content of the article is summarized in the introduction.

Random ergodic theorems and real cocycles

AbstractWe study mean convergence of ergodic averages

Powers of sequences and convergence of ergodic averages

\frac{1}{N}\sum\nolimits_{n = 0}^{N - 1} {f^\circ \tau ^{k_n (\omega )} ( * )}

On the filtering problem for stationary random /spl Zopf//sup 2/-fields

associated to a measure-preserving transformation or flow τ along the random sequence of times κn(ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT.We prove that the sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations.When no assumption is made onF, the random sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (kn(ω)) is not almost surely universally good for the class of weakly mixing transformations.