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

Pinelis, Iosif

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 412-426.

It is proved that the best constant factor in the Rademacher-gaussian tail comparison is between two explicitly defined absolute constants $c_1$ and $c_2$ such that $c_2$$\approx$1.01 $c_1$. A discussion of relative merits of this result versus limit theorems is given.

Classification : 60E15, 62G10, 62G15, 60G50, 62G35
Mots clés : probability inequalities, Rademacher random variables, sums of independent random variables, Student's test, self-normalized sums

[1] V. Bentkus, A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand. Lithuanian Math. J. 42 (2002) 262-269. | Zbl

[2] V. Bentkus, An inequality for tail probabilities of martingales with differences bounded from one side. J. Theoret. Probab. 16 (2003) 161-173 | Zbl

[3] V. Bentkus, On Hoeffding's inequalities. Ann. Probab. 32 (2004) 1650-1673. | Zbl

[4] S.G. Bobkov, F. Götze, C. Houdré, On Gaussian and Bernoulli covariance representations. Bernoulli 7 (2002) 439-451. | Zbl

[5] G.E. Collins, Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lect. Notes Comput. Sci. 33 (1975) 134-183. | Zbl

[6] M.L. Eaton, A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2 (1974) 609-614. | Zbl

[7] D. Edelman, An inequality of optimal order for the tail probabilities of the $T$ statistic under symmetry. J. Amer. Statist. Assoc. 85 (1990) 120-122. | Zbl

[8] B. Efron, Student’s $t$ test under symmetry conditions. J. Amer. Statist. Assoc. 64 (1969) 1278-1302. | Zbl

[9] S.E. Graversen, G. Peškir, Extremal problems in the maximal inequalities of Khintchine. Math. Proc. Cambridge Philos. Soc. 123 (1998) 169-177. | Zbl

[10] S. Łojasiewicz, Sur les ensembles semi-analytiques. Actes du Congrès International des Mathématiciens (Nice, 1970). Tome 2, Gauthier-Villars, Paris (1970) 237-241. | Zbl

[11] I. Pinelis, Extremal probabilistic problems and Hotelling’s $T^2$ test under a symmetry condition. Ann. Statist. 22 (1994) 357-368. | Zbl

[12] I. Pinelis, Optimal tail comparison based on comparison of moments. High dimensional probability (Oberwolfach, 1996). Birkhäuser, Basel Progr. Probab. . 43 (1998) 297-314. | Zbl

[13] 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. | Zbl

[14] I. Pinelis, On exact maximal Khinchine inequalities. High dimensional probability, II (Seattle, WA, 1999). Birkhäuser Boston, Boston, MA Progr. Probab.. 47 (2000) 49-63. | Zbl

[15] 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). | Zbl

[16] 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. | MR | Zbl

[17] I. Pinelis, On normal domination of (super)martingales. Electronic Journal of Probality 11 (2006) 1049-1070. | MR | Zbl

[18] I. Pinelis, On l'Hospital-type rules for monotonicity. J. Inequal. Pure Appl. Math. 7 (2006) art40 (electronic). | Zbl

[19] 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. | MR | Zbl

[20] I. Pinelis, On inequalities for sums of bounded random variables. Preprint (2006) http://arxiv.org/abs/math.PR/0603030.

[21] A.A. Tarski, A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica, Calif. (1948). | MR | Zbl