• Peng, Zuoxiang ;
  • Nadarajah, Saralees
  • 투고 : 2010.11.20
  • 발행 : 2012.05.31


Let $X_1$, $X_2$,${\ldots}$, $X_n$ be a sequence of independent and identically distributed random variables with common distribution function $F$. Convergence rates for the moments of extremes are studied by virtue of second order regularly conditions. A unified treatment is also considered under second order von Mises conditions. Some examples are given to illustrate the results.


convergence rate of moments;maximum;second order regular variation;second order von Mises condition


  1. S. H. Cheng and C. G. Jiang, The Edgeworth expansion for distributions of extreme values, Sci. China Ser. A 44 (2001), no. 4, 427-437.
  2. L. de Haan and A. Ferreira, Extreme Value Theory: An Introduction, Springer-Verlag, New York, 2006.
  3. L. de Haan and S. I. Resnick, Second order regular variation and rates of convergence in extreme value theory, Ann. Probab. 24 (1996), no. 1, 97-124.
  4. L. de Haan and U. Stadtmuller, Generalized regular variation of second order, J. Austral. Math. Soc. Ser. A 61 (1996), no. 3, 381-395.
  5. J. L. Geluk, On the domain of attraction of exp(exp(-x)), Statist. Probab. Lett. 31 (1996), no. 2, 91-95.
  6. J. L. Geluk and L. de Haan, Regular variation, Extensions and Tauberian Theorems, CWI Tract, 40. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.
  7. M. I. Gomes and L. de Haan, Approximation by penultimate extreme value distributions, Extremes 2 (1999), no. 1, 71-85.
  8. P. Hall, On the rate of convergence of normal extremes, J. Appl. Probab. 16 (1979), no. 2, 433-439.
  9. W. J. Hall and J. A. Wellner, The rate of convergence in law of the maximum of an exponential sample, Statist. Neerlandica 33 (1979), no. 3, 151-154.
  10. E. Kaufmann, Penultimate approximations in extreme value theory, Extremes 3 (2000), no. 1, 39-55.
  11. K. Liu and Z. Peng, The convergence rates of conditional moments, Journal of Southwest University (Natural Science) 29 (2007), 5-8.
  12. Z. Peng, M. Liu, and S. Nadarajah, Conditions based on conditional moment for maxstable limit laws, Extremes 11 (2008), no. 4, 329-337.
  13. J. Pickands, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881-889.
  14. S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer-Verlag, New York, 1987.