A variance bound for a general function of independent noncommutative random variables

Abstract

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab. 44(5) (2016), 3431{3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.

Mathematics Subject Classification (2010): Primary 46L53; Secondary 60E15.

Keywords: Efron-Stein inequality, random matrix, noncommutative probability, trace, conditional expectation