Florence Merlevède

Recent advances in invariance principles for stationary sequences

In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.

Bernstein inequality and moderate deviations under strong mixing conditions

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

Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains

We consider a large class of piecewise expanding maps T of [0; 1] with a neutral xed point, and their associated Markov chain Yi whose transition kernel is the PerronFrobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f T i satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Yi) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f T i may belong to the domain of normal attraction of a stable law of index p2 (1; 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

AbstractIn this paper we study the behavior of sums of a linear process

On martingale approximations and the quenched weak invariance principle

X_k = \sum {_{j = - \infty }^\infty } a_j (\xi _{k - j} )

Invariance principles for linear processes with application to isotonic regression

associated to a strictly stationary sequence

On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

\{ \xi _k \} _{k \in \mathbb{Z}}