Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extremal properties of Rademacher functions
with applications to the Khintchine
and Rosenthal inequalities

Authors: T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtman and J. Zinn
Journal: Trans. Amer. Math. Soc. 349 (1997), 997-1027
MSC (1991): Primary 60E15, 60G50; Secondary 26D07, 46E30
MathSciNet review: 1390980
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the $p$ moment of a sum of independent symmetric random variables to that of the $p$ and $2$ moments of the individual variables, are computed in the range $2<p\le 4$. This complements the work of Utev who has done the same for $p>4$. The qualitative nature of the extreme cases turns out to be different for $p<4$ than for $p>4$. The method developed yields results in some more general and other related moment inequalities.

References [Enhancements On Off] (What's this?)

  • [E1] Morris L. Eaton, A note on symmetric Bernoulli random variables, Ann. Math. Statist. 41 (1970), 1223–1226. MR 0268930
  • [E2] M. R. Eaton, A probability inequality for linear combinations of bounded random variables, Ann. Statist. 2 (1974), 609-613.
  • [H] Uffe Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1981), no. 3, 231–283 (1982). MR 654838
  • [JSZ] W. B. Johnson, G. Schechtman, and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13 (1985), no. 1, 234–253. MR 770640
  • [K] Ryszard Komorowski, On the best possible constants in the Khintchine inequality for 𝑝≥3, Bull. London Math. Soc. 20 (1988), no. 1, 73–75. MR 916079, 10.1112/blms/20.1.73
  • [KS] Stanisław Kwapień and Jerzy Szulga, Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, Ann. Probab. 19 (1991), no. 1, 369–379. MR 1085342
  • [MO] 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-London, 1979. MR 552278
  • [Pe] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR 0388499
  • [Ph] Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
  • [P] Iosif Pinelis, Extremal probabilistic problems and Hotelling’s 𝑇² test under a symmetry condition, Ann. Statist. 22 (1994), no. 1, 357–368. MR 1272088, 10.1214/aos/1176325373
  • [PU] I. F. Pinelis and S. A. Utev, Estimates of moments of sums of independent random variables, Teor. Veroyatnost. i Primenen. 29 (1984), no. 3, 554–557 (Russian). MR 761144
  • [R] 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. MR 0271721
  • [S] S. B. Stečkin, On best lacunary systems of functions, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 357–366 (Russian). MR 0131097
  • [Sz] S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), no. 2, 197–208. MR 0430667
  • [T] Michel Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17 (1989), no. 4, 1546–1570. MR 1048946
  • [U1] S. A. Utev, Extremal problems in moment inequalities, Limit theorems of probability theory, Trudy Inst. Mat., vol. 5, “Nauka” Sibirsk. Otdel., Novosibirsk, 1985, pp. 56–75, 175 (Russian). MR 821753
  • [U2] S. A. Utev, Extremal problems in moment inequalities, Limit Theorems in Probability Theory, Trudy Inst. Math., Novosibirsk, 1985, pp. 56-75 (in Russian).
  • [W] P. Whittle, Bounds for the moments of linear and quadratic forms in independent variables, Teor. Verojatnost. i Primenen. 5 (1960), 331–335 (English, with Russian summary). MR 0133849
  • [Y] R. M. G. Young, On the best possible constants in the Khintchine inequality, J. London Math. Soc. (2) 14 (1976), no. 3, 496–504. MR 0438089

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 60E15, 60G50, 26D07, 46E30

Retrieve articles in all journals with MSC (1991): 60E15, 60G50, 26D07, 46E30

Additional Information

T. Figiel
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Abrahama 18, 81–825 Sopot, Poland

P. Hitczenko
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695–8205

W. B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

G. Schechtman
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel

J. Zinn
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Khintchine inequality, Rosenthal inequality, Orlicz function, extremal problem, Rademacher functions
Received by editor(s): December 22, 1994
Additional Notes: The first, second and fourth authors were participants in the NSF Workshop in Linear Analysis & Probability, Texas A&M University
Professors Hitczenko, Johnson, and Zinn were supported in part by NSF grants
Johnson, Schechtman and Zinn were supported in part by US–Israel Binational Science Foundation
Article copyright: © Copyright 1997 American Mathematical Society