Abstract
We establish several Schur-convexity type results under fixed variance for weighted sums of independent gamma random variables and obtain nonasymptotic bounds on their Rényi entropies. In particular, this pertains to the recent results by Bartczak–Nayar–Zwara as well as Bobkov–Naumov–Ulyanov, offering simple proofs of the former and extending the latter.
Similar content being viewed by others
Data Availability
No associated data.
References
Ball, K.: Cube slicing in \(R^n\). Proc. Amer. Math. Soc. 97(3), 465–473 (1986)
Bartczak, M., Nayar, P., Zwara, S.: Sharp variance-entropy comparison for nonnegative Gaussian quadratic forms. IEEE Trans. Inform. Theory 67(12), 7740–7751 (2021)
Bhatia, R.: Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, (1997)
Białobrzeski, M., Nayar, P.: Rényi entropy and variance comparison for symmetric log-concave random variables, preprint (2021), arXiv:2108.10100
Bobkov, S.G., Chistyakov, G.P.: On concentration functions of random variables. J. Theoret. Probab. 28(3), 976–988 (2015)
Bobkov, S.G., Chistyakov, G.P.: Entropy power inequality for the Rényi entropy. IEEE Trans. Inform. Theory 61(2), 708–714 (2015)
Bobkov, S.G., Marsiglieti, A., Melbourne, J.: Concentration functions and entropy bounds for discrete log-concave distributions. Comb. Probab. Comput. 31, 54–72 (2022)
Bobkov, S.G., Marsiglietti, A.: Variants of the entropy power inequality. IEEE Trans. Inform. Theory 63(12), 7747–7752 (2017)
Bobkov, S. G., Naumov, A. A., Ulyanov, V. V.: Two-sided inequalities for the density function’s maximum of weighted sum of chi-square variables, preprint (2020) arXiv:2012.10747. In: Recent Developments in Stochastic Methods and Applications. Springer Proceedings in Mathematics & Statistics, Vol. 371 , pp. 178–189. Springer, Cham (2021)
Chasapis, G., Gurushankar, K., Tkocz, T.: Sharp bounds on \(p\)-norms for sums of independent uniform random variables, \(0<p<1\), preprint (2021), arXiv:2105.14079. To appear in J. Anal. Math
Chasapis, G., König, H., Tkocz, T.: From Ball’s cube slicing inequality to Khinchin-type inequalities for negative moments. J. Funct. Anal. 281(9), 109185 (2021)
Chasapis, G., Nayar, P., Tkocz, T.: Slicing \(\ell _p\)-balls reloaded: stability, planar sections in \(\ell _1\), preprint (2021), arXiv:2109.05645, to appear in Ann. Probab
Eskenazis, A., Nayar, P., Tkocz, T.: Gaussian mixtures: entropy and geometric inequalities. Ann. Prob. 46(5), 2908–2945 (2018)
Eskenazis, A., Nayar, P., Tkocz, T.: Sharp comparison of moments and the log-concave moment problem. Adv. Math. 334, 389–416 (2018)
Feller, W.: An introduction to probability theory and its applications, vol. II. Second edition Wiley Inc, New York-London-Sydney (1971)
Fradelizi, M.: Hyperplane sections of convex bodies in isotropic position. Beiträge Algebra Geom. 40(1), 163–183 (1999)
Hensley, D.: Slicing the cube in \(R^n\) and probability (bounds for the measure of a central cube slice in \(R^n\) by probability methods). Proc. Amer. Math. Soc. 73(1), 95–100 (1979)
Koldobsky, A.: An application of the Fourier transform to sections of star bodies. Israel J. Math. 106, 157–164 (1998)
Koldobsky, A.: Fourier analysis in convex geometry. Mathematical Surveys and Monographs, Vol. 116, American Mathematical Society, Providence, RI, (2005)
Koldobsky, A., Ryabogin, D., Zvavitch, A.: Fourier analytic methods in the study of projections and sections of convex bodies, In: Fourier analysis and convexity, pp. 119–130 , Boston, MA, (2004)
König, H., Koldobsky, A.: On the maximal measure of sections of the n-cube. geometric analysis, mathematical relativity, and nonlinear partial differential equations. Contemp. Math 599, 123–155 (2012)
König, H., Koldobsky, A.: On the maximal perimeter of sections of the cube. Adv. Math. 346, 773–804 (2019)
Li, J.: Rényi entropy power inequality and a reverse. Studia Math. 242(3), 303–319 (2018)
Li, J., Marsiglietti, A., Melbourne, J.: Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities, In: Geometric aspects of functional analysis, pp. 95-123, Springer, Cham (2020)
Livshyts, G., Paouris, G., Pivovarov, P.: On sharp bounds for marginal densities of product measures. Israel J. Math. 216(2), 877–889 (2016)
Madiman, M., Melbourne, J., Xu, P.: Forward and reverse entropy power inequalities in convex geometry, In: Convexity and concentration pp. 427-485, Springer, New York, NY, (2017)
Madiman, M., Nayar, P., Tkocz, T.: Sharp moment-entropy inequalities for symmetric log-concave distributions. IEEE Trans. Inform. Theory 67(1), 81–94 (2021)
Marsiglietti, A., Melbourne, J.: On the entropy power inequality for the Rényi entropy of order \([0, 1]\). IEEE Trans. Inform. Theory 65(3), 1387–1396 (2019)
Melbourne, J., C. Roberto, C.: Quantitative form of Ball’s Cube slicing in Rn and equality cases in the min-entropy power inequality, preprint (2021), arXiv:2109.03946
Melbourne, J., C. Roberto, C.: Transport-majorization to analytic and geometric inequalities, preprint (2021), arXiv:2110.03641
Melbourne, J., Tkocz, T.: Reversals of Rényi entropy inequalities under log-concavity. IEEE Trans. Inform. Theory 67(1), 45–51 (2021)
Moriguti, S.: A lower bound for a probability moment of any absolutely continuous distribution with finite variance. Ann. Math. Statistics 23, 286–289 (1952)
Nguyen, H. H., Vu, V. H.: (. Small ball probability, inverse theorems, and applications. In Erdös centennial, pp. 409-463. Springer, Berlin, Heidelberg (2013)
Ram, E., Sason, I.: On Rényi entropy power inequalities. IEEE Trans. Inform. Theory 62(12), 6800–6815 (2016)
Rényi, A.: On measures of entropy and information, In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, Vol. I. University California Press, Berkeley, Calif., pp. 547–561 1961
Yu, Y.: On an inequality of Karlin and Rinott concerning weighted sums of i.i.d. random variables. Adv. Appl. Prob. 40(4), 1223–1226 (2008)
Acknowledgements
We are indebted to Han Nguyen for many fruitful and illuminating discussions. We should like to thank an anonymous referee very much for their insightful comments greatly improving this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest.
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
TT’s research supported in part by NSF grant DMS-1955175.
Rights and permissions
About this article
Cite this article
Chasapis, G., Singh, S. & Tkocz, T. Entropies of Sums of Independent Gamma Random Variables. J Theor Probab 36, 1227–1242 (2023). https://doi.org/10.1007/s10959-022-01192-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-022-01192-y