Concentration Inequalities for Semi-bounded Martingales

In this paper we extend the results of de la Peña [3]. The main method that we use is the theory of decoupling, which has been developed in de la Peña [2] and [3]. Decoupling theory provides a general framework for analyzing problems involving dependent random variables as if they were independent. We will apply the theory of decoupling as in de la Peña [3] and some new inequalities for independent random variables motivated by Maurer [1] to obtaining extensions of exponential inequalities of de la Peña [3]. The paper is organized as follows. In Section 2, by usual methods, we present some new exponential inequalities for random variables and discrete time martingales. A brief introduction to the theory of decoupling is presented in Section 3. In Section 4, we use the theory of decoupling to establish some further results which refine those in Section 2 and extend de la Peña’s exponential inequalities from bounded martingale difference sequence to semi-bounded ones.