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Exponential inequalities for martingales with applications. (English) Zbl 1320.60058

Summary: The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of G. Bennett [J. Am. Stat. Assoc. 57, No. 297, 33–45 (1962; Zbl 0104.11905)], D. A. Freedman [Ann. Probab. 3, 100–118 (1975; Zbl 0313.60037)], V. H. de la Peña [Ann. Probab. 27, No. 1, 537–564 (1999; Zbl 0942.60004)], I. Pinelis [Ann. Probab. 22, No. 4, 1679–1706 (1994; Zbl 0836.60015)] and S. van de Geer [Ann. Stat. 23, No. 5, 1779–1801 (1995; Zbl 0852.60019); in: Empirical process techniques for dependent data. Boston, MA: Birkhäuser. 161–169 (2002; Zbl 1027.60013)]. 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 also provided.

MSC:

60E15 Inequalities; stochastic orderings
60G42 Martingales with discrete parameter
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