×

zbMATH — the first resource for mathematics

Berry-Esseen bounds in the entropic central limit theorem. (English) Zbl 1307.60011
Summary: Berry-Esseen-type bounds for total variation and relative entropy distances to the normal law are established for the sums of non-i.i.d. random variables.

MSC:
60F05 Central limit and other weak theorems
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Artstein, S., Ball, K.M., Barthe, F., Naor, A.: On the rate of convergence in the entropic central limit theorem. Probab. Theory Relat. Fields 129(3), 381-390 (2004) · Zbl 1055.94004
[2] Barron, AR, Entropy and the central limit theorem, Ann. Probab., 14, 336-342, (1986) · Zbl 0599.60024
[3] Barron, AR; Johnson, O, Fisher information inequalities and the central limit theorem, Probab. Theory Relat. Fields, 129, 391-409, (2004) · Zbl 1047.62005
[4] Bhattacharya, R.N., Ranga Rao, R.: Normal Approximation and Asymptotic Expansions. Wiley, New York (1976). Also: Soc. for Industrial and Appl. Math., Philadelphia (2010) · Zbl 0331.41023
[5] Bobkov, S.G., Chistyakov, G.P., Götze, F.: Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem. Ann. Probab. ArXiv:1104.3994 v1 [math.PR] (2011) · Zbl 1175.60020
[6] Bobkov, S.G., Chistyakov, G.P., Götze, F.: Bounds for characteristic functions in terms of quantiles and entropy. Electron. Commun. Probab. 17 (2012), paper no. 21, electronic
[7] Bobkov, SG; Götze, F, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163, 1-28, (1999) · Zbl 0924.46027
[8] Brown, L.D.: A Proof of the Central Limit Theorem Motivated by the Cramer-Rao Inequality. Statistics And Probability: Essays in Honor of C. R. Rao, pp. 141-148. North-Holland, Amsterdam (1982)
[9] Carlen, EA; Soffer, A, Entropy production by block variable summation and central limit theorems, Comm. Math. Phys., 140, 339-371, (1991) · Zbl 0734.60024
[10] Cover, TM; Dembo, A; Thomas, JA, Information-theoretic inequalities, IEEE Trans. Inf. Theory, 37, 1501-1518, (1991) · Zbl 0741.94001
[11] Csiszár, I, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hung., 2, 299-318, (1967) · Zbl 0157.25802
[12] Esseen, C-G, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math., 77, 1-125, (1945) · Zbl 0060.28705
[13] Fedotov, AA; Harremoës, P; Topsøe, F, Refinements of pinsker’s inequality, IEEE Trans. Inf. Theory, 49, 1491-1498, (2003) · Zbl 1063.94017
[14] Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971) · Zbl 0219.60003
[15] Ibragimov, I.A., Linnik, J.V.: Independent and Stationarily Connected Variables. Izdat. “Nauka”, Moscow (1965) · Zbl 0154.42201
[16] Johnson, O.: Information Theory and the Central Limit Theorem. Imperial College Press, London (2004) · Zbl 1061.60019
[17] Kullback, S, A lower bound for discrimination in terms of variation, IEEE Trans. Inf. Theory, T-13, 126-127, (1967)
[18] Linnik, JV, An information-theoretic proof of the central limit theorem with the lindeberg condition, Theory Probab. Appl., 4, 288-299, (1959) · Zbl 0097.13103
[19] Petrov, V.V.: Sums of independent random variables. Springer, Berlin (1975) · Zbl 0322.60043
[20] Pinelis, IF; Utev, SA, Estimates of moments of sums of independent random variables, Theory Probab. Appl., 29, 574-577, (1984) · Zbl 0566.60017
[21] Pinsker, M.S.: Information and information stability of random variables and processes. Translated and edited by Amiel Feinstein Holden-Day, Inc., San Francisco (1964) · Zbl 0125.09202
[22] Prohorov, YV, A local theorem for densities (Russian), Doklady Akad. Nauk SSSR (N.S.), 83, 797-800, (1952) · Zbl 0046.35301
[23] Rio, E, Upper bounds for minimal distances in the central limit theorem, Ann. Inst. Henri Poincaré Probab. Stat., 45, 802-817, (2009) · Zbl 1175.60020
[24] Rosenthal, HP, On the subspaces of \(L^{p}\)\((p{\>}2)\) spanned by sequences of independent random variables, Isr. J. Math., 8, 273-303, (1970) · Zbl 0213.19303
[25] Senatov, V.V.: Central Limit Theorem. Exactness of Approximation and Asymptotic Expansions (Russian). TVP Science Publishers, Moscow (2009)
[26] Statulevičius, VA, Limit theorems for densities and the asymptotic expansions for distributions of sums of independent random variables, Theory Probab. Appl., 10, 682-695, (1965) · Zbl 0212.22802
[27] Sirazhdinov, SH; Mamatov, M, On Mean convergence for densities, Theory Probab. Appl., 7, 424-428, (1962) · Zbl 0302.60015
[28] Talagrand, M, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6, 587-600, (1996) · Zbl 0859.46030
[29] Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003) · Zbl 1106.90001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.