Some exponential inequalities for acceptable random variables and complete convergence

Abstract

Some exponential inequalities for a sequence of acceptable random variables are obtained, such as Bernstein-type inequality, Hoeffding-type inequality. The Bernsteintype inequality for acceptable random variables generalizes and improves the corresponding results presented by Yang for NA random variables and Wang et al. for NOD random variables. Using the exponential inequalities, we further study the complete convergence for acceptable random variables. MSC (2000): 60E15, 60F15.

General Theorems on Exponential and Rosenthal's Inequalities

“…If (2.1) is satisfied for all λ ∈ R then we obtain the notion of acceptable r.v. 's ( [1], [7]). If (2.1) is true for η 1 , η 2 , .…”

“…Then it was extended to dependent sequences. Our next theorem is a version of Theorem 2.3 in [7], where acceptable r.v. 's were studied.…”

“…For acceptable r.v. 's an appropriate version of the original proof is described in [7] (see the proof of Theorem 2.3 in [7]).…”

“…We shall find the maximal version of Hoeffding's inequality. In [7] the maximal Hoeffding's inequality for acceptable r.v. 's was not considered.…”

“…Exponential inequalities were obtained for acceptable sequences, furthermore some versions of the notion of acceptability were introduced, see e.g. [7] and [10].…”