Rocky Mountain Journal of Mathematics

Inequalities for sums of random variables in noncommutative probability spaces

Ghadir Sadeghi and Mohammad Sal Moslehian

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

Abstract

In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq 2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak {M}, \tau )$ be a noncommutative probability space, $\mathfrak {N}$ be a von Neumann subalgebra of $\mathfrak {M}$ with the corresponding conditional expectation $\mathcal {E}_{\mathfrak {N}}$ and let subalgebras $\mathfrak {N}\subseteq \mathfrak {A}_j\subseteq \mathfrak {M}\,\,(j=1, \cdots , n)$ be successively independent over $\mathfrak {N}$. Let $x_j\in \mathfrak {A}_j$ be self-adjoint such that $a_j\leq x_j\leq b_j$ for some real numbers $a_j\lt b_j$ and $\mathcal {E}_{\mathfrak {N}}(x_j)=\mu $ for some $\mu \geq 0$ and all $1\leq j\leq n$. Then for any $t>o$ it holds that \[ {\rm Prob}\bigg (\bigg |\sum _{j=1}^n x_j-n\mu \bigg |\geq t\bigg )\leq 2 \exp \bigg \{\frac {-2t^2}{\sum _{j=1}^n(b_j-a_j)^2}\bigg \}. \]

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 1 (2016), 309-323.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1464035865

Digital Object Identifier
doi:10.1216/RMJ-2016-46-1-309

Mathematical Reviews number (MathSciNet)
MR3506091

Zentralblatt MATH identifier
1357.46057

Subjects
Primary: 46L53: Noncommutative probability and statistics 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B11: Probability theory on linear topological spaces [See also 28C20]

Keywords
Noncommutative probability space von Neumann algebra noncommutative Rosenthal inequality noncommutative Bennett inequality noncommutative Hoeffding inequality

Citation

Sadeghi, Ghadir; Moslehian, Mohammad Sal. Inequalities for sums of random variables in noncommutative probability spaces. Rocky Mountain J. Math. 46 (2016), no. 1, 309--323. doi:10.1216/RMJ-2016-46-1-309. https://projecteuclid.org/euclid.rmjm/1464035865


Export citation

References

  • G. Bennett, Probability inequalities for the sum of independent random variables, J. Amer. Stat. Assoc. 57 (1962), 33–45.
  • V. Bentkus, On Hoeffding's inequalities, Ann. Prob. 32 (2004), 1650–1673.
  • D.L. Burkholder, Distribution function inequalities for martingales, Ann. Prob. 1 (1973), 19–42.
  • J.B. Conway, A course in functional analysis, Grad. Texts Math., Springer-Verlag, New York, 1990.
  • S. Dirksen, B. de Pagter, D. Potapov and F. Sukochev, Rosenthal inequalities in noncommutative symmetric spaces, J. Funct. Anal. 261 (2011), 2890–2925.
  • W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Stat. Assoc. 58 (1963), 13–30.
  • X.-L. Hu, An extension of Rosenthal's inequality, Appl. Math. Comp. 218 (2011), 4638–4640.
  • W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Prob. 13 (1985), 234–253.
  • M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities, II, Applications, Israel J. Math. 167 (2008), 227–282.
  • M. Junge and Q. Zeng, Noncommutative Bennett and Rosenthal inequalities, Ann. Prob. 41 (2013), 4287–4316.
  • M.S. Moslehian, M. Tominaga and K.-S. Saito, Schatten $p$-norm inequalities related to an extended operator parallelogram law, Linear Alg. Appl. 435 (2011), 823–829.
  • E. Nelson, Note on noncommutative integration, J. Funct. Anal. 15 (1974), 103–116.
  • B. de Pagter, Noncommutative Banach function spaces, Positivity, 197-227, Birkhüser, Basel, 2007.
  • H.P. Rosenthal, On the subspaces of $L_p (p > 2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303.
  • M.B. Ruskai, Inequalities for traces on von Neumann algebras, Comm. Math. Phys. 26 (1972), 280–289.
  • Gh. Sadeghi, Non-commutative Orlicz spaces associated to a modular on $\tau$-measurable operators, J. Math. Anal. Appl. 395 (2012), 705–715.
  • J.A. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comp. Math. 12 (2012), 389–434.
  • Q. Xu, Embedding of $C_q$ and $R_q$ into noncommutative $L_p$-spaces, $1\leq p<q \leq 2$, Math. Ann. 335 (2006), 109–131.
  • ––––, Operator spaces and noncommutative $L_p$, Lect. Summer School Banach Spaces Operator Spaces, Nankai University, China, 2007.
  • F.J. Yeadon, Noncommutative $L^p$-spaces, Proc. Camb. Phil. Soc. 77 (1975), 91–102.