Sublinear bounds on the maximum of the density for sums of independent random variables are given in terms of the maxima of the densities of summands.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Dembo, T. M. Cover, and J. A. Thomas, “Information-theoretic inequalities,” IEEE Trans. Inform. Theory, 37, 1501–1518 (1991).
O. Johnson, Information Theory and the Central Limit Theorem, Imperial College Press, London (2004).
S. G. Bobkov and M. Madiman, “Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures,” J. Funct. Anal., 262, 3309–3339 (2012).
A. J. Stam, “Some inequalities satisfied by the quantities of information of Fisher and Shannon,” Information Control, 2, 101–112 (1959).
V. V. Petrov, Sums of Independent Random Values [in Russian], Moscow (1972).
A. N. Shiryaev, Probability [in Russian], Moscow (1980).
S. G. Bobkov, G. P. Chistyakov, and F. Götze, “Bounds for characteristic functions in terms of quantiles and entropy,” Election. Commun. Probab., 17, (2012).
E. H. Lieb, “Proof of an entropy conjecture of Wehrl,” Comm. Math Phys., 62, 35–41 (1978).
W. Beckner, “Inequalities in Fourier analysis,” Ann Math., 2, 159–182 (1975).
H. J. Brascamp and E. H. Lieb, “Best constants in Young’s inequality, its converse, and its generalization to more than three functions,” Advances Math., 20, 151–173 (1976).
F. Barthe, “Optimal Young’s inequality and its converse: a simple proof,” Geom. Funct. Anal., 8, 234–242 (1998).
S. G. Bobkov, C. Houdre, and P. Tetali, “The subgaussian constant and concentration inequalities,” Israel J. Math., 156, 255–283 (2006).
B. A. Rogozin, “An estimate for the maximum of the convolution for bounded densities,” Teor. Veroyatn. Primen., 32, 53–61 (1987).
Yu. V. Prokhovor, “Extremal problems in limit theorems,” Proc. Sixth All-Union Conf. Theory Prob. and Math. Statist. (Vilnius, 1960), Vilnius (1962), pp. 77–84.
K. Ball, “Cube slicing in R n,” Proc. Amer. Math. Soc., 97 465–473 (1986).
K. Ball, “Some remarks on the geometry of convex sets,” Lect. Notes Math., 1317, 224–231 (1988).
L. P. Postnikova and A. A. Yudin, “ A sharpened form of an inequality for the concentration function,” Teor. Veroyatn. Primen., 23, 376–379 (1978).
A. L. Miroshnikov and B. A. Rogozin, “ Inequalities for the concentration function,” Teor. Veroyatn. Primen., 25, 178–183 (1980).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 408, 2012, pp. 62–73.
Rights and permissions
About this article
Cite this article
Bobkov, S.G., Chistyakov, G.P. Bounds on the Maximum of the Density for Sums of Independent Random Variables. J Math Sci 199, 100–106 (2014). https://doi.org/10.1007/s10958-014-1836-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1836-9