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Toward the best constant factor for the Rademacher-Gaussian tail comparison

Published online by Cambridge University Press:  17 August 2007

Iosif Pinelis
Iosif Pinelis*
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931 USA; ipinelis@mtu.edu
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Abstract

It is proved that the best constant factor in the Rademacher-Gaussian tail comparison is between two explicitly defined absolute constants c1 and c2 such that c2≈1.01 c1. A discussion of relative merits of this result versus limit theorems is given.

Keywords

Probability inequalitiesRademacher random variablessums of independent random variablesStudent's testself-normalized sums.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 412 - 426
Copyright
© EDP Sciences, SMAI, 2007

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References

Bentkus, V., A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand. Lithuanian Math. J. 42 (2002) 262–269. CrossRef
Bentkus, V., An inequality for tail probabilities of martingales with differences bounded from one side. J. Theoret. Probab. 16 (2003) 161–173 CrossRef
Bentkus, V., Hoeffding's, On inequalities. Ann. Probab. 32 (2004) 1650–1673. CrossRef
Bobkov, S.G., Götze, F., Houdré, C., Gaussian, On and Bernoulli covariance representations. Bernoulli 7 (2002) 439–451. CrossRef
Collins, G.E., Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lect. Notes Comput. Sci. 33 (1975) 134–183. CrossRef
Eaton, M.L., A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2 (1974) 609–614. CrossRef
Edelman, D., An inequality of optimal order for the tail probabilities of the T statistic under symmetry. J. Amer. Statist. Assoc. 85 (1990) 120–122.
Efron, B., Student's t test under symmetry conditions. J. Amer. Statist. Assoc. 64 (1969) 1278–1302.
Graversen, S.E., Peškir, G., Extremal problems in the maximal inequalities of Khintchine. Math. Proc. Cambridge Philos. Soc. 123 (1998) 169–177. CrossRef
S. Łojasiewicz, Sur les ensembles semi-analytiques. Actes du Congrès International des Mathématiciens (Nice, 1970). Tome 2, Gauthier-Villars, Paris (1970) 237–241.
Pinelis, I., Extremal probabilistic problems and Hotelling's T2 test under a symmetry condition. Ann. Statist. 22 (1994) 357–368. CrossRef
I. Pinelis, Optimal tail comparison based on comparison of moments. High dimensional probability (Oberwolfach, 1996). Birkhäuser, Basel Progr. Probab. . 43 (1998) 297–314.
I. Pinelis, Fractional sums and integrals of r-concave tails and applications to comparison probability inequalities Advances in stochastic inequalities (Atlanta, GA, 1997). Amer. Math. Soc., Providence, RI. 234 Contemp. Math., . (1999) 149–168.
I. Pinelis, On exact maximal Khinchine inequalities. High dimensional probability, II (Seattle, WA, 1999). Birkhäuser Boston, Boston, MA Progr. Probab.. 47 (2000) 49–63.
I. Pinelis, Birkhäuser, Basel L'Hospital type rules for monotonicity: applications to probability inequalities for sums of bounded random variables. J. Inequal. Pure Appl. Math. 3 (2002) Article 7, 9 pp. (electronic).
I. Pinelis, Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above. IMS Lecture Notes Monograph Series 51 (2006) 33-52.
I. Pinelis, On normal domination of (super)martingales. Electronic Journal of Probality 11 (2006) 1049-1070.
I. Pinelis, On l'Hospital-type rules for monotonicity. J. Inequal. Pure Appl. Math. 7 (2006) art40 (electronic).
I. Pinelis, Exact inequalities for sums of asymmetric random variables, with applications. Probability Theory and Related Fields (2007) DOI 10.1007/s00440-007-0055-4.
I. Pinelis, On inequalities for sums of bounded random variables. Preprint (2006) http://arxiv.org/abs/math.PR/0603030.
A.A. Tarski, A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica, Calif. (1948).