A comment on rates of convergence for density function in extreme value theory and Rényi entropy

Author:
Ali Saeb

Journal:
Theor. Probability and Math. Statist. **108** (2023), 169-183

MSC (2020):
Primary 60F10

DOI:
https://doi.org/10.1090/tpms/1191

Published electronically:
May 2, 2023

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References |
Similar Articles |
Additional Information

Abstract: De Haan and Resnick [Ann. Probab. 10 (1982), no. 2, 396–413] have shown that the Rényi entropy of order $\beta$ ($\beta >1$) of normalized sample maximum of independent and identically distributed (iid) random variables with continuous differentiable density converges to the Rényi entropy of order $\beta$ 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.

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References
- A. Aksomaitis and A. Joidmaitis,
*Convergence rate for density of maximum of independent random variabes*, Lithuanian Mathematical Journal **37** (1997), no. 2.
- S. Artstein, K. M. Ball, F. Barthe, and A. Naor,
*On the rate of convergence in the entropic central limit theorem*, Probability Theory Related Fields **129** (2004), no. 3, 381–390. MR **2128238**
- A. R. Barron,
*Entropy and the central limit theorem*, Ann. Probab. **14** (1986), no. 1, 336–342. MR **815975**
- F. Buryak and Y. Mishura,
*Convexity and robustness of the Rényi entropy*, Mod. Stoch. Theory Appl. **8** (2021), no. 3, 387–412. MR **4312787**
- Hongfei C. and Yiming D.,
*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**
- L. de Haan and S. I. Resnick,
*Local limit theorems for sample extremes*, Ann. Probab. **10** (1982), no. 2, 396–413. MR **647512**
- —,
*Second-order regular variation and rates of convergence in extreme value theory*, Ann. Probab. **24** (1996), no. 1, 97–124. MR **1387628**
- P. Embrechts, C. Klüppelberg, and T. Mikosch,
*Modeling extremal events for insurance and finance*, vol. 33, Springer-Verlag, Berlin, 1997. MR **1458613**
- J. Galambos,
*The asymptotic theory of extreme order statistics*, second ed., Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1987. MR **936631**
- S. Ghosal, J. K. Ghosh, and A. W. van der Vaart,
*Convergence rates of posterior distributions*, Ann. Statist. **28** (2000), no. 2, 500–531. MR **1790007**
- E. T. Jaynes,
*Information theory and statistical mechanics*, Phys. Rev. (2) **106** (1957), 620–630. MR **87305**
- —,
*Prior probabilities*, IEEE Trans. Syst., Man. Cybern., (SSC-4) (1968), 227–241.
- O. Johnson,
*Information theory and the central limit theorem*, Imperial College Press, 2004. MR **2109042**
- O. Johnson and A. Barron,
*Fisher information inequalities and the central limit theorem*, Probab. Theory Related Fields **129** (2004), no. 3, 391–409. MR **2128239**
- O. Johnson and C. Vignat,
*Some results concerning maximum Rényi entropy distributions*, Ann. Inst. H. Poincaré Probab. Statist. **43** (2007), no. 3, 339–351. MR **2319701**
- J. V. Linnik,
*An information theoretic proof of the central limit theorem with Lindeberg conditions*, Theor. Probability Appl. **4** (1959), 288–299. MR **124081**
- E. Omey,
*Rates of convergence for densities in extreme value theory*, Ann. Probab. **16** (1988), no. 2, 479–486. MR **929058**
- S. Ravi and A. Saeb,
*On convergence of entropy of distribution functions in the max domain of attraction of max stable laws*, arXiv (2014). MR **3962245**
- A. Renyi,
*On measures of entropy and information*, (1961), 547–561. MR **0132570**
- S. 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**
- A. Saeb,
*On relative Rényi entropy convergence in max domain of attraction*, Yokohama Math. J. **64** (2018), 83–98. MR **3962245**
- X. Shen and L. Wasserman,
*Rates of convergence of posterior distributions*, Ann. Statist. **29** (2001), no. 3, 687–714. MR **1865337**
- R. L. Smith,
*Uniform rates of convergence in extreme value theory*, Adv. in Appl. Probab. **14** (1982), no. 3, 600–622. MR **665296**
- T. J. Sweeting,
*On domains of uniform local attraction in extreme value theory*, Ann. Probab. **13** (1985), no. 1, 196–205. MR **770637**

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Additional Information

**Ali Saeb**

Affiliation:
Department of Economic Sciences, Indian Institute of Science Education and Research, Bhopal 462 066, India

Email:
ali.saeb@gmail.com

Keywords:
Rate of convergence,
Rényi entropy,
densities convergence,
max stable laws,
max domain of attraction

Received by editor(s):
May 4, 2022

Accepted for publication:
October 4, 2022

Published electronically:
May 2, 2023

Additional Notes:
The first version of the paper was written while I was visiting the Institute for Mathematical Sciences, National University of Singapore. The visit was supported by the Institute.

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv