Acceptable random variables in non-commutative probability spaces

A class of random variables called negatively orthant dependent random variables defined in the classical setting of probability spaces by Lehman in 1966. Joag-Dev and Proschan extended this class of random variables and showed every sequence of negatively associated random variables is negatively orthant dependent. In 2008, acceptable random variables are defined by Antonini in the classical setting of probability spaces. Let {Xn}n∈N be a sequence of random variables in a probability space (Ω,F,P), where Ω is a sample space, F is a σ-algebra of the subsets of  Ω, and P is a probability measure in F. Furthermore let E be the expectation of random variables in (Ω,F,P). In the proof of limit theorems it is so important to obtain an exponential bound for a partial sum i=1n(Xi-EXi) . Sung et. al., obtained an exponential bound for i=1n(Xi-EXi) and proved it for the cacceptable random variables class. Kim, Nooghabi and Azarnoosh, Roussas and Xing, obtained exponential bound for negatively associated random variables. In this paper, we define a class of acceptable random variables in noncommutative (quantum) probability spaces. In fact this paper transfers some of probability inequalities from the commutative probability spaces into the noncommutative probability spaces.

In this scheme, first for the convenience of the reader we repeat the main definitions of noncommutative probability spaces. In the second step we define acceptable random variables in the noncommutative probability spaces and prove some probability inequalities for this class of random variables.

In general case, probability inequalities determine upper and lower bounds for the expectation of a random variable or the probability measure of an event. Sometimes we can not obtain the expectation or the probability measure exactly. In these situations, these bounds are important for control the expectation of a random variable or the probability measure of an event.In this paper we want to answer this question:Can we obtain these bounds for probability inequalities in von Neumann algebras?
Methods are used in this paper to compare classical setting are different. One of difficulties of the noncommutative setting is this fact that the product of two positive elements is not necessarily positive element in a C*-algebra; Because it is not self-adjoint. Another one is this fact that there is not guarantee for existence of the maximum of two positive operators.

The following conclusions were drawn from this research. The probability inequalities are proved for acceptable random variables in Noncommutative probability spaces. They are a generalization of probability inequalities for negatively orthant dependent random variables in probability theory. We show under what situations, sequence 1ni=1nxin≥1is completely convergence, where {xn}n≥1 is a sequence of noncommutative self-adjoint random variables. We show under what situations, sequence 1nβi=1nxin≥1for every β>1 is completely convergence, where {xn}n≥1 is a sequence of noncommutative self-adjoint random variables. All the results of this paper can be used for random matrices.