On Hoeffding and Bernstein type inequalities for sums of random variables in non-additive measure spaces and complete convergence

Working with real phenomena, one often faces situations where additivity assumption is unavailable. Non-additive measures and Choquet integral are attracting much attention from scientists in many different areas such as financial economics, economic modelling, probability theory and statistics. Hoeffding’s and Bernstein’s inequalities are two powerful tools that can be applied in many studies of the asymptotic behaviour of inference problems in probability theory, model selection, stochastic processes and economic modelling. One thing that seems missing is the developments of Hoeffding’s and Bernstein’s inequalities for sums of random variables in non-additive cases. The purposes of this paper are to extend Hoeffding’s and Bernstein’s inequalities for sums of random variables from probability measure space to non-additive measure space, and then establish two complete convergence theorems for more general form.