Exponential inequalities for martingales with applications

Abstract

International audienceThe paper is devoted to establishing some general exponential inequalities for super-martingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of de la Peña's inequality to self-normalized deviations is also provided

Bernstein type inequalities for self-normalized martingales with applications

“…The following technical lemma is from Fan et al [13]. For reader's convenience, we shall give a proof following [13]. Then (U i (λ), F i ) i=0,··· ,n is a supermartingale, and satisfies that for all λ ∈ [0, 1),…”

New insights on concentration inequalities for self-normalized martingales

“…Over the last two decades, there has been a renewed interest in this area of probability. To be more precise, extensive studies have been made in order to establish concentration inequalities for (M n ) without boundedness assumptions on its increments [5], [14], [16], [17], [27], [28]. For example, it was established in [5] that for any positive x and y, (1.1) P(|M n | x, [M] n + <M> n y) 2 exp − x 2 2y .…”

Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

“…An extension of the Hoeffding-Azuma inequalities for the weighted sum of uniformly bounded martingale differences can be found in [45]. Generalizations of the exponential inequalities for the case of real-valued supermartingales were obtained in [25] and recently generalized in [23], where the authors use change of probability measure techniques, and give applications for estimation in the general parametric (real-valued) autoregressive model. Extensions of [25] for the case of supermartingales in Banach spaces were obtained in [41].…”