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mmr 2022, 8(2): 117-128 |
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Sadeghi G, Divandar M S. Acceptable random variables in non-commutative probability spaces. mmr 2022; 8 (2) :117-128
URL: http://mmr.khu.ac.ir/article-1-2979-en.html
URL: http://mmr.khu.ac.ir/article-1-2979-en.html
1- Hakim Sabzevari University , g.sadeghi@hsu.ac.ir
2- Hakim Sabzevari University
2- Hakim Sabzevari University
Abstract: (1129 Views)
Introduction
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{ X n } n i =1 n ( X i - X i i =1 n ( X i - X i
Material and methods
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.
Results and discussion
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 aC *
Conclusion
The following conclusions were drawn from this research.
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
Material and methods
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.
Results and discussion
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
Conclusion
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
- We show under what situations, sequence
- All the results of this paper can be used for random matrices.
Type of Study: S |
Subject:
alg
Received: 2019/07/15 | Revised: 2022/11/16 | Accepted: 2020/10/11 | Published: 2022/05/21 | ePublished: 2022/05/21
Received: 2019/07/15 | Revised: 2022/11/16 | Accepted: 2020/10/11 | Published: 2022/05/21 | ePublished: 2022/05/21
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