Theory of Probability and Mathematical Statistics
A comment on rates of convergence for density function in extreme value theory and Rényi entropy
Ali Saeb
Link
Abstract: De Haan and Resnick [Ann. Probab. 10 (1982), no. 2, 396–413] have shown that the Rényi entropy of order β(β>1) of normalized sample maximum of independent and identically distributed (iid) random variables with continuous differentiable density converges to the Rényi entropy of order β of a max stable law. In this paper, we review the rate of convergence for density function in extreme value theory. Finally, we study the rate of convergence for Rényi entropy in the case of normalized sample maxima.
Keywords: Rate of convergence, Rényi entropy, densities convergence, max stable laws, max domain of attraction
Bibliography: A. Aksomaitis and A. Joidmaitis, Convergence rate for density of maximum of independent random variabes, Lithuanian Mathematical Journal 37 (1997), no. 2.
Shiri Artstein, Keith M. Ball, Franck Barthe, and Assaf Naor, On the rate of convergence in the entropic central limit theorem, Probab. Theory Related Fields 129 (2004), no. 3, 381–390. MR 2128238, DOI 10.1007/s00440-003-0329-4
Andrew R. Barron, Entropy and the central limit theorem, Ann. Probab. 14 (1986), no. 1, 336–342. MR 815975
Filipp Buryak and Yuliya Mishura, Convexity and robustness of the Rényi entropy, Mod. Stoch. Theory Appl. 8 (2021), no. 3, 387–412. MR 4312787, DOI 10.15559/21-vmsta185
Hongfei Cui and Yiming Ding, The convergence of the Rényi entropy of the normalized sums of IID random variables, Statist. Probab. Lett. 80 (2010), no. 15-16, 1167–1173. MR 2657479, DOI 10.1016/j.spl.2010.03.012
L. de Haan and S. I. Resnick, Local limit theorems for sample extremes, Ann. Probab. 10 (1982), no. 2, 396–413. MR 647512, DOI 10.1214/aop/1176993865
Laurens de Haan and Sidney Resnick, Second-order regular variation and rates of convergence in extreme-value theory, Ann. Probab. 24 (1996), no. 1, 97–124. MR 1387628, DOI 10.1214/aop/1042644709
Paul Embrechts, Claudia Klüppelberg, and Thomas Mikosch, Modelling extremal events, Applications of Mathematics (New York), vol. 33, Springer-Verlag, Berlin, 1997. For insurance and finance. MR 1458613, DOI 10.1007/978-3-642-33483-2
Janos Galambos, The asymptotic theory of extreme order statistics, 2nd ed., Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1987. MR 936631
Subhashis Ghosal, Jayanta K. Ghosh, and Aad W. van der Vaart, Convergence rates of posterior distributions, Ann. Statist. 28 (2000), no. 2, 500–531. MR 1790007, DOI 10.1214/aos/1016218228
E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. (2) 106 (1957), 620–630. MR 87305, DOI 10.1103/PhysRev.106.620
—, Prior probabilities, IEEE Trans. Syst., Man. Cybern., (SSC-4) (1968), 227–241.
Oliver Johnson, Information theory and the central limit theorem, Imperial College Press, London, 2004. MR 2109042, DOI 10.1142/9781860945373
Oliver Johnson and Andrew Barron, Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields 129 (2004), no. 3, 391–409. MR 2128239, DOI 10.1007/s00440-004-0344-0
Oliver Johnson and Christophe Vignat, Some results concerning maximum Rényi entropy distributions, Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 3, 339–351 (English, with English and French summaries). MR 2319701, DOI 10.1016/j.anihpb.2006.05.001
Ju. V. Linnik, An information-theoretic proof of the central limit theorem with Lindeberg conditions, Theor. Probability Appl. 4 (1959), 288–299 (Russian, with English summary). MR 124081, DOI 10.1137/1104028
E. Omey, Rates of convergence for densities in extreme value theory, Ann. Probab. 16 (1988), no. 2, 479–486. MR 929058, DOI 10.1214/aop/1176991768
Ali Saeb, On relative Rényi entropy convergence of the max domain of attraction, Yokohama Math. J. 64 (2018), 83–98. MR 3962245
Alfréd Rényi, On measures of entropy and information, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I, Univ. California Press, Berkeley, Calif., 1961, pp. 547–561. MR 0132570
Sidney I. Resnick, Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4, Springer-Verlag, New York, 1987. MR 900810, DOI 10.1007/978-0-387-75953-1
Ali Saeb, On relative Rényi entropy convergence of the max domain of attraction, Yokohama Math. J. 64 (2018), 83–98. MR 3962245
Xiaotong Shen and Larry Wasserman, Rates of convergence of posterior distributions, Ann. Statist. 29 (2001), no. 3, 687–714. MR 1865337, DOI 10.1214/aos/1009210686
Richard L. Smith, Uniform rates of convergence in extreme-value theory, Adv. in Appl. Probab. 14 (1982), no. 3, 600–622. MR 665296, DOI 10.2307/1426676
T. J. Sweeting, On domains of uniform local attraction in extreme value theory, Ann. Probab. 13 (1985), no. 1, 196–205. MR 770637, DOI 10.1214/aop/1176993075
