Empirical invariance principle for ergodic torus automorphisms; genericity. - Archive ouverte HAL Access content directly
Journal Articles Stochastics and Dynamics Year : 2008

Empirical invariance principle for ergodic torus automorphisms; genericity.

Olivier Durieu

Abstract

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.
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Dates and versions

hal-00322351 , version 1 (17-09-2008)

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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⟩
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