Empirical invariance principle for ergodic torus automorphisms; genericity.

Olivier Durieu , Philippe Jouan (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

We consider the dynamical system given by an algebraic ergodic automorphism $T$ on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process $(f\circ T^i)_{i\in\N}$, where $f$ is a given $\R$-valued function. We give a sufficient condition on $f$ for this Central Limit Theorem to hold. In a second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least 3 and Hölder continuous if the dimension is 1 or 2. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

Mots clés: Empirical process; Partially hyperbolic dynamical system; Functional central limit theorem; Multiple mixing; Distribution function; Morse function; Genericity.; Genericity

Domaines: Probabilités [math.PR]

Olivier Durieu, Philippe Jouan. Empirical invariance principle for ergodic torus automorphisms; genericity.. Stochastics and Dynamics, 2008, 8 (2), pp.173-195. ⟨10.1142/S0219493708002287⟩. ⟨hal-00322351⟩