## Rocky Mountain Journal of Mathematics

### Inequalities for sums of random variables in noncommutative probability spaces

#### 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

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

#### 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

#### 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.