## Electronic Journal of Probability

### Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Iosif Pinelis

#### Abstract

Exact upper bounds on the generalized moments $\operatorname{\mathsf {E}} f(S_n)$ of sums $S_n$ of independent nonnegative random variables $X_i$ for certain classes $\mathcal{F}$ of nonincreasing functions $f$ are given in terms of (the sums of) the first two moments of the $X_i$’s. These bounds are of the form $\operatorname{\mathsf {E}} f(\eta )$, where the random variable $\eta$ is either binomial or Poisson depending on whether $n$ is fixed or not. The classes $\mathcal{F}$ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $\operatorname{\mathsf {P}} (S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $\operatorname{\mathsf {E}} \exp \{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 20, 19 pp.

Dates
Accepted: 25 February 2016
First available in Project Euclid: 11 March 2016

https://projecteuclid.org/euclid.ejp/1457706456

Digital Object Identifier
doi:10.1214/16-EJP4474

Mathematical Reviews number (MathSciNet)
MR3485362

Zentralblatt MATH identifier
1338.60061

#### Citation

Pinelis, Iosif. Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables. Electron. J. Probab. 21 (2016), paper no. 20, 19 pp. doi:10.1214/16-EJP4474. https://projecteuclid.org/euclid.ejp/1457706456

#### References

• [1] Bengt von Bahr and Carl-Gustav Esseen, Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$, Ann. Math. Statist 36 (1965), 299–303.
• [2] George Bennett, Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc. 57 (1962), no. 297, 33–45.
• [3] V. Bentkus, N. Kalosha, and M. van Zuijlen, On domination of tail probabilities of (super)martingales: explicit bounds, Liet. Mat. Rink. 46 (2006), no. 1, 3–54.
• [4] Vidmantas Bentkus, On Hoeffding’s inequalities, Ann. Probab. 32 (2004), no. 2, 1650–1673.
• [5] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968.
• [6] Stéphane Boucheron, Olivier Bousquet, Gábor Lugosi, and Pascal Massart, Moment inequalities for functions of independent random variables, Ann. Probab. 33 (2005), no. 2, 514–560.
• [7] George E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Quantifier elimination and cylindrical algebraic decomposition (Linz, 1993), Texts Monogr. Symbol. Comput., Springer, Vienna, 1998, pp. 85–121.
• [8] Christopher R. Dance, An inequality for the sum of independent bounded random variables, J. Theoret. Probab. 27 (2014), no. 2, 358–369.
• [9] Morris L. Eaton, A note on symmetric Bernoulli random variables, Ann. Math. Statist. 41 (1970), 1223–1226.
• [10] Morris L. Eaton, A probability inequality for linear combinations of bounded random variables, Ann. Statist. 2 (1974), 609–613.
• [11] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.
• [12] R. Ibragimov and Sh. Sharakhmetov, The best constant in the Rosenthal inequality for nonnegative random variables, Statist. Probab. Lett. 55 (2001), no. 4, 367–376.
• [13] Rafał Latała, Estimation of moments of sums of independent real random variables, Ann. Probab. 25 (1997), no. 3, 1502–1513.
• [14] S. Łojasiewicz, Sur les ensembles semi-analytiques, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 237–241.
• [15] Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979.
• [16] Andreas Maurer, A bound on the deviation probability for sums of non-negative random variables, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Article 15, 6 pp. (electronic).
• [17] Vicente Muñoz and Ulf Persson, Interviews with three Fields medallists: Andrei Okounkov, Newsletter of the European Mathematical Society (2006, December), no. 62, 34–35, http://www.ams.org/notices/200703/comm-fields-interviews.pdf, pp. 1–2.
• [18] Andrew Odlyzko, Review: Experimental Mathematics in Action, Amer. Math. Monthly 118 (2011), no. 10, 946–951, http://dx.doi.org/10.4169/amer.math.monthly.118.10.946.
• [19] Adam Osȩkowski, Sharp inequalities for sums of nonnegative random variables and for a martingale conditional square function, ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010), 243–256.
• [20] I. Pinelis, Convex cones of generalized multiply monotone functions and the dual cones, ArXiv e-prints (2016), arXiv:1501.06599v2 [math.CA], to appear in Banach Journal of Mathematical Analysis.
• [21] I. F. Pinelis and A. I. Sakhanenko, Remarks on inequalities for probabilities of large deviations, Theory Probab. Appl. 30 (1985), no. 1, 143–148.
• [22] I. F. Pinelis and S. A. Utev, Sharp exponential estimates for sums of independent random variables, Theory Probab. Appl. 34 (1989), no. 2, 340–346.
• [23] Iosif Pinelis, Extremal probabilistic problems and Hotelling’s $T^2$ test under a symmetry condition, Ann. Statist. 22 (1994), no. 1, 357–368.
• [24] Iosif Pinelis, Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. 22 (1994), no. 4, 1679–1706.
• [25] Iosif Pinelis, Optimal tail comparison based on comparison of moments, High dimensional probability (Oberwolfach, 1996), Progr. Probab., vol. 43, Birkhäuser, Basel, 1998, pp. 297–314.
• [26] Iosif Pinelis, Fractional sums and integrals of $r$-concave tails and applications to comparison probability inequalities, Advances in stochastic inequalities (Atlanta, GA, 1997), Contemp. Math., vol. 234, Amer. Math. Soc., Providence, RI, 1999, pp. 149–168.
• [27] Iosif Pinelis, Exact inequalities for sums of asymmetric random variables, with applications, Probab. Theory Related Fields 139 (2007), no. 3–4, 605–635.
• [28] Iosif Pinelis, On the Bennett-Hoeffding inequality, http://arxiv.org/abs/0902.4058; a shorter version appeared in [31], 2009.
• [29] Iosif Pinelis, Positive-part moments via the Fourier–Laplace transform, J. Theor. Probab. 24 (2011), 409–421.
• [30] Iosif Pinelis, Exact Rosenthal-type inequalities for $p=3$, and related results, Statistics & Probability Letters 83 (2013), no. 12, 2634–2637.
• [31] Iosif Pinelis, On the Bennett–Hoeffding inequality, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 50 (2014), no. 1, 15–27.
• [32] Iosif Pinelis, Best possible bounds of the von Bahr–Esseen type, Ann. Funct. Anal. 6 (2015), no. 4, 1–29.
• [33] Iosif Pinelis, Exact binomial, Poisson, and Gaussian bounds for the left tails of sums of nonnegative random variables, Version 3, http://arxiv.org/abs/1503.06482, 2015.
• [34] Iosif Pinelis, Exact Rosenthal-type bounds, Ann. Probab. 43 (2015), no. 5, 2511–2544.
• [35] Julie Rehmeyer, Voevodsky’s Mathematical Revolution, Scientific American (2013), no. October 1, http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/
• [36] Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303.
• [37] L. R. Shenton, Inequalities for the normal integral including a new continued fraction, Biometrika 41 (1954), 177–189.
• [38] Alfred Tarski, A Decision Method for Elementary Algebra and Geometry, RAND Corporation, Santa Monica, Calif., 1948.
• [39] I. S. Tyurin, Some optimal bounds in the central limit theorem using zero biasing, Statist. Probab. Lett. 82 (2012), no. 3, 514–518.