The Michigan Mathematical Journal

Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces

Ali Talebi, Mohammad Sal Moslehian, and Ghadir Sadeghi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if x1,x2,,xn are successively independent self-adjoint random variables in a noncommutative probability space (M,τ) such that τ(xk)=0 and sksk1=sk1sk, where sk=j=1kxj, then, for any λ>0, there exists a projection e such that

1(λ+max 1knxk)2k=1nvar(xk)τ(e)τ(sn2)λ2. As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.

Article information

Michigan Math. J., Volume 68, Issue 1 (2019), 57-69.

Received: 1 February 2017
Revised: 25 September 2017
First available in Project Euclid: 8 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L10: General theory of von Neumann algebras 47A30: Norms (inequalities, more than one norm, etc.) 60F99: None of the above, but in this section


Talebi, Ali; Moslehian, Mohammad Sal; Sadeghi, Ghadir. Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces. Michigan Math. J. 68 (2019), no. 1, 57--69. doi:10.1307/mmj/1541667627.

Export citation


  • [1] C. J. K. Batty, The strong law of large numbers for states and traces of a $W^{\ast}$-algebra, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 177–191.
  • [2] I. Cuculescu, Martingales on von Neumann algebras, J. Multivariate Anal. 1 (1971), 17–27.
  • [3] N. Etemadi, On some classical results in probability theory, Sankhya, Ser. A 47 (1985), no. 2.
  • [4] A. Gut, Probability: a graduate course, Second edition, Springer Texts Statist., Springer, New York, 2013.
  • [5] J. Hajék and A. Renyi, Generalization of an inequality of Kolmogorov, Acta Math. Acad. Sci. Hung. 6 (1955), no. 3–4, 281–283.
  • [6] M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. II. Applications, Israel J. Math. 167 (2008), 227–282.
  • [7] M. Junge and Q. Zeng, Noncommutative Bennett and Rosenthal inequalities, Ann. Probab. 41 (2013), no. 6, 4287–4316.
  • [8] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103–116.
  • [9] N. Randrianantoanina, A weak type inequality for noncommutative martingales and applications, Proc. Lond. Math. Soc. 91 (2005), no. 3, 509–544.
  • [10] Gh. Sadeghi and M. S. Moslehian, Noncommutative martingale concentration inequalities, Illinois J. Math. 58 (2014), no. 2, 561–575.
  • [11] Gh. Sadeghi and M. S. Moslehian, Inequalities for sums of random variables in noncommutative probability spaces, Rocky Mountain J. Math. 46 (2016), no. 1, 309–323.
  • [12] A. Talebi, M. S. Moslehian, and Gh. Sadeghi, Noncommutative Blackwell–Ross martingale inequality, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (to appear), arXiv:1705.07122.
  • [13] Q. Xu, Operator spaces and noncommutative $L_{p}$-spaces, Lectures in the Summer School on Banach Spaces and Operator Spaces, Nankai University, China, 2007.