Olivier Durieu

Invariance principles for self-similar set-indexed random fields

For a stationary random field

Approximating class approach for empirical processes of dependent sequences indexed by functions

(X_j)_{j\in\Z^d}

From infinite urn schemes to decompositions of self-similar Gaussian processes

and some measure m on

A sequential empirical CLT for multiple mixing processes with application to

\R^d

\mathcal{B}

, we consider the set-indexed weighted sum process

-geometrically ergodic Markov chains

S_n(A)=\sum_{j\in\Z^d}m(nA\cap R_j)^\frac12 X_j

EMPIRICAL INVARIANCE PRINCIPLE FOR ERGODIC TORUS AUTOMORPHISMS: GENERICITY

, where R_j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X_js and an entropy condition on the class of sets A. The limit processes are self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsovs type representations to choose appropriate measures m and particular sets A, we show that these limits can be Levy (fractional) Brownian fields or (fractional) Brownian sheets.

On Sums of Indicator Functions in Dynamical Systems

We study weak convergence of empirical processes of dependent data

Invariance principles for operator-scaling Gaussian random fields

(X_i)_{i\geq 0}

AN INDICATOR FUNCTION LIMIT THEOREM IN DYNAMICAL SYSTEMS

, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class