On concentration, noise and entropy estimation in dynamical systems

By

Cesar MALDONADO 

(21 septembre 2012)

On concentration, noise and entropy estimation in dynamical systems.pdf

Abstract

This thesis is divided into three parts. In the rst part we brie
y describe the class
of dynamical systems considered. We also give some known results on the study of

uctuations of observables in dynamical systems such as the central limit theorem, large
deviations and concentration inequalities.
In the second part we study dynamical systems perturbed by observational noise. We
prove that if a dynamical system satis es a concentration inequality then the system with
observational noise also satis es a concentration inequality. We apply these inequalities
to obtain
uctuation bounds for the auto-covariance function, the empirical measure,
the kernel density estimator and the correlation dimension. Next, we study the work of
S. Lalley on the problem of signal recovery. Given a time series of a chaotic dynamical
system with observational noise, one can e ectively eliminate the noise in average by
using Lalley's algorithm. A chapter of this thesis is devoted to the proof of consistency
of that algorithm. We end up the second part with a numerical quest for the best
parameters of Lalley's algorithm.
The third part is devoted to entropy estimation in one-dimensional Gibbs measures.
We study the
uctuations of two entropy estimators. The rst one is based on the empirical
frequencies of observation of typical blocks. The second is based on the time a typical
orbit takes to hit an independent typical block. We apply concentration inequalities to
obtain bounds on the
uctuation of these estimators.