In these notes, we first give a brief overwiew of martingales methods, from Paul Lévy (1935) untill now, to explain why these methods have become a central tool in probability, statistics and ergodic theory. Next, we present some recent results for/or based on martingales: exponential bounds for super-martingales, concentration inequalities for Lipschitz functionals of dynamical systems, oracle inequalities for the Cox model in a high dimensional setting, and invariance principles for stationary sequences.
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Chazottes et al. (2014) studied this question.
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