Abstract
The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pena, 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 Pena’s inequality to self-normalized deviations is also provided.
Highlights
Assume that we are given a sequence of real-valued supermartingale differencesi=0,...,n defined on some probability space (Ω, F, P), where ξ0 = 0 and {∅, Ω} =
The following exponential inequality for supermartingales can be found in Freedman [16]
Following the work of Freedman [16], Shorack and Wellner [34], van de Geer [35] and de la Peña [8], we develop some new methods, based on changes of probability measure, for establishing some general exponential inequalities for supermartingales
Summary
The following exponential inequality for supermartingales can be found in Freedman [16]. Following the work of Freedman [16], Shorack and Wellner [34], van de Geer [35] and de la Peña [8], we develop some new methods, based on changes of probability measure, for establishing some general exponential inequalities for supermartingales. In Theorem 2.1, we obtain two exponential inequalities for supermartingales under a very general condition. Assume that E(eλξi |Fi−1) ≤ exp{f (λ)Vi−1} for some λ ∈ (0, ∞), for a positive function f (λ) and for some Fi−1-measurable random variables Vi−1. If Vi−1 = E(ξi2|Fi−1) is the conditional variance, inequality (1.17) reduces to Theorem. Inequality (1.17) implies the following result, where Vi−1 is not the conditional variance. The proofs of the theorems and their corollaries are in the same sections
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