Abstract
We derive Khinchine type inequalities for even moments with optimal constants from the result of Walkup (J Appl Probab 13:76–85, 1976) which states that the class of log-concave sequences is closed under the binomial convolution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baernstein, II, A., Culverhouse R.C.: Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions. Stud. Math 152, 231–248 (2002)
Barlow R.E., Marshall A.W., Proschan F.: Properties of probability distributions with monotone hazard rate. Ann. Math. Stat. 34, 375–389 (1963)
Borell C.: Complements of Lyapunov’s inequality. Math. Ann. 205, 323–331 (1973)
Czerwiński, W.: Khinchine inequalities (in Polish). University of Warsaw, Master thesis (2008)
Gurvits, L.: A short proof, based on mixed volumes, of Liggett’s theorem on the convolution of ultra-logconcave sequences. Electron. J. Combin. 16, Note 5 (2009)
Haagerup U.: The best constants in the Khintchine inequality. Stud. Math. 70, 231–283 (1982)
Johnson O.: Log-concavity and the maximum entropy property of the Poisson distribution. Stoch. Process. Appl. 117, 791–802 (2007)
Khintchine A.: Über dyadische Brüche. Math. Z. 18, 109–116 (1923)
König H., Kwapień S.: Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors. Positivity 5, 115–152 (2001)
Kwapień S., Latała R., Oleszkiewicz K.: Comparison of moments of sums of independent random variables and differential inequalities. J. Funct. Anal. 136, 258–268 (1996)
Liggett T.M.: Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A 79, 315–325 (1997)
Latała R., Oleszkiewicz K.: On the best constant in the Khinchin–Kahane inequality. Studia Math. 109, 101–104 (1994)
Oleszkiewicz, K.: Comparison of moments via Poincaré-type inequality. In: Advances in Stochastic Inequalities (Atlanta, GA, 1997), Contemp. Math. 234. American Mathematical Society, Providence 135–148 (1999)
Szarek S.: On the best constant in the Khintchine inequality. Stud. Math. 58, 197–208 (1976)
Walkup D.W.: Pólya sequences, binomial convolution and the union of random sets. J. Appl. Probab. 13, 76–85 (1976)
Whittle P.: Bounds for the moments of linear and quadratic forms in independent random variables. Theory Probab. Appl. 5, 302–305 (1960)
Acknowledgments
We are grateful to Matthieu Fradelizi and Olivier Guédon for pointing to us the article of Walkup, and for their help in tracing some other references.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of K. Oleszkiewicz was partially supported by Polish MNiSzW Grant N N201 397437.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Nayar, P., Oleszkiewicz, K. Khinchine type inequalities with optimal constants via ultra log-concavity. Positivity 16, 359–371 (2012). https://doi.org/10.1007/s11117-011-0130-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-011-0130-z