Lithuanian Mathematical Journal

, Volume 54, Issue 4, pp 403–408 | Cite as

A note on random signs

  • Dainius Dzindzalieta


Let ε, ε1, ε2, . . . be independent identically distributed Rademacher random variables, P{ε = ± 1} = 1/2. Denote S = ∑ i = 1 a i ε i , where a1, a2, . . . is a (weight) sequence of nonrandom real numbers such that ∑ i = 1 a i 2  ≤ 1. We prove that \( \mathrm{P}\left\{S\ge x\right\}\le 1/4+\left(1-\sqrt{2-2/{x}^2}\right)/8 \) for \( x\in \left(1,\sqrt{2}\right] \).


weighted sums of Rademacher random variables random signs bounds for tail probabilities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Ben-Tal and A. Nemirovski, On safe tractable approximations of chance-constrained linear matrix inequalities, Math. Oper. Res., 34(1):1–25, 2009.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    A. Ben-Tal, A. Nemirovski, and C. Roos, Robust solutions of uncertain quadratic and conic-quadratic problems, SIAM J. Optim., 13(2):535–560, 2002.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    V.Yu. Bentkus, On large deviations in Banach spaces, Teor. Veroyatn. Primen., 31(4):710–716, 1986 (in Russian).MATHMathSciNetGoogle Scholar
  4. 4.
    V. Bentkus, An inequality for large deviation probabilities of sums of bounded i.i.d. random variables, Lith. Math. J., 41(2):112–119, 2001.CrossRefMathSciNetGoogle Scholar
  5. 5.
    V. Bentkus, An inequality for tail probabilities of martingales with bounded differences, Lith. Math. J., 42(3):255–261, 2002.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    V. Bentkus, On measure concentration for separately Lipschitz functions in product spaces, Isr. J. Math., 158:1–17, 2007.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    V. Bentkus and D. Dzindzalieta, A tight Gaussian bound for weighted sums of Rademacher random variables, Bernoulli, 2014 (forthcomming).Google Scholar
  8. 8.
    V. Bentkus and T. Juškeviˇcius, Bounds for tail probabilities of martingales using skewness and kurtosis, Lith. Math. J., 48(1):30–37, 2008.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    K. Derinkuyu and M.C. Pınar, On the S-procedure and some variants, Math. Methods Oper. Res., 64(1):55–77, 2006.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    K. Derinkuyu, M.C. Pınar, and A. Camcı, An improved probability bound for the approximate S-Lemma, Oper. Res. Lett., 35(6):743–746, 2007.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    B. Efron, Student’s t-test under symmetry conditions, J. Am. Stat. Assoc., 64(328):1278–1302, 1969.MATHMathSciNetGoogle Scholar
  12. 12.
    R.K. Guy, Any answers anent these analytical enigmas?, Am. Math. Mon., 93(4):279–281, 1986.CrossRefMathSciNetGoogle Scholar
  13. 13.
    S. He, Z.-Q. Luo, J. Nie, and S. Zhang, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization, SIAM J. Optim., 19(2):503–523, 2008.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    J.-B. Hiriart-Urruty, A new series of conjectures and open questions in optimization and matrix analysis, ESAIM, Control Optim. Calc. Var., 15(2):454–470, 2009.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc., 58:13–30, 1963.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    R. Holzman and D.J. Kleitman, On the product of sign vectors and unit vectors, Combinatorica, 12(3):303–316, 1992.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Math. Program., Ser. B, 109(2–3):283–317, 2007.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    A.M.-C. So, Improved approximation bound for quadratic optimization problems with orthogonality constraints, in Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (New York, January 4–6, 2009), SIAM, Philadelphia, PA, 2009, pp. 1201–1209.Google Scholar
  19. 19.
    I. Tyurin, New estimates of the convergence rate in the Lyapunov theorem, preprint, 2009, arXiv:0912.0726.Google Scholar
  20. 20.
    A.V. Zhubr, On one extremal problem for n-cube, Tr. Komi Nauchn. Tsentra UrO Ross. AN, 2(10):4–11, 2012 (in Russian).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania

Personalised recommendations