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Illustration de l'actualité : PhD defense Gabriela Ciolek : Bootstrap and uniform bounds for Harris Markov chains

PhD defense Gabriela Ciolek : Bootstrap and uniform bounds for Harris Markov chains

vendredi
14
décembre
2018

PhD Comics : I'm defending my thesis, Mom !

14;00, B310

Jury

  • M. Patrice Bertail, Université Paris Nanterre, PhD co-advisor
  • M. Stephan Clémençon, Télécom ParisTech, PhD co-advisor
  • M. Randal Douc, Télécom SudParis, Examiner  
  • Mme. Cécile Durot, Université Paris Nanterre, Examiner
  • M. Kengo Kato, Cornell University, Reviewer
  • Mme. Sana Louhichi, Université de Grenoble, Examiner
  • M. Olivier Wintenberger, Sorbonne Université, Reviewer 

Abstract

This thesis concentrates on some extensions of empirical processes theory when the data are Markovian. More specifically, we focus on some developments of bootstrap, robustness and statistical learning theory in a Harris recurrent framework. Our approach relies on the regenerative methods that boil down to division of sample paths of the regenerative Markov chain under study into independent and identically distributed (i.i.d.) blocks of observations. In the first part of the thesis we derive uniform bootstrap central limit theorems for Harris recurrent Markov chains. We use the aforementioned results to obtain uniform bootstrap central limit theorems for Fréchet differentiable functionals of Harris Markov chains. Propelled by vast applications, we discuss how to extend some concepts of robustness from the i.i.d. framework to a Markovian setting. In particular, we consider the case when the data are Piecewise-determinic Markov processes. Next, we propose the residual and wild bootstrap procedures for periodically autoregressive processes and show their consistency. In the second part of the thesis we establish maximal versions of Bernstein, Hoeffding and polynomial tail type concentration inequalities. We obtain the inequalities as a function of covering numbers and moments of time returns and blocks. Finally, we use those tail inequalities to derive generalization bounds for minimum volume set estimation for regenerative Markov chains.