Skip to main content

Introduction to Gnedenko (1943) On the Limiting Distribution of the Maximum Term in a Random Series

  • Chapter
Breakthroughs in Statistics

Part of the book series: Springer Series in Statistics ((PSS))

Abstract

Gnedenko’s paper was the first mathematically rigorous treatment of the fundamental limit theorems of extreme value theory. In its influence on the probabilistic theory of extremes, the paper set the agenda for the next 30 years. Its influence on statistical methodology was less direct, although many of the more recent statistical developments have exploited the probabilistic theory that Gnedenko’s paper spawned.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Anderson, C.W. (1970). Extreme value theory for a class of discrete distributions with applications to some stochastic processes, J. Appl. Prob., 7, 99–113.

    Article  MATH  Google Scholar 

  • Balkema, A.A., and de Haan, L. (1972). On R. von Mises’ condition for the domain of attraction of exp(-e-x), Ann. Math. Statist., 43, 1352–1354.

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. (1961). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables, Math. Scand., 9, 383–394.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. (1963). On the limit behaviour of extreme order statistics, Ann. Math. Statist., 34, 992–1002.

    Article  MathSciNet  MATH  Google Scholar 

  • Berman, S.M. (1962). Limiting distribution of the maximum term in a sequence of dependent random variables, Ann. Math. Statist., 33, 894–908.

    Article  MathSciNet  MATH  Google Scholar 

  • Berman, S.M. (1964). Limit theorems for the maximum term in stationary sequences, Ann. Math. Statist, 35, 502–516.

    Article  MathSciNet  MATH  Google Scholar 

  • Bingham, N.H., Goldie, CM., and Teugels, J.L. (1987). Regular Variation. Cambridge University Press, New York.

    MATH  Google Scholar 

  • Clough, D.J., and Kotz, S. (1965). Extreme-value distributions with a special queueing model application, CORS J., 3, 96–109.

    MathSciNet  Google Scholar 

  • Davison, A.C., and Smith, R.L. (1989). Models for excedances over high thresholds (with discussion), J. Roy. Statist. Soc., Ser. B, 52, 393–442.

    MathSciNet  Google Scholar 

  • de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts No. 32, Amsterdam.

    Google Scholar 

  • de Haan, L. (1971). A form of regular variation and its application to the domain of attraction of the double exponential, Z. Wahrsch. v. Geb., 17, 241–258.

    Article  MATH  Google Scholar 

  • de Haan, L. (1976). Sample extremes: An elementary introduction, Statist. Neerlan-dica, 30, 161–172.

    Article  MATH  Google Scholar 

  • de Haan, L., and Resnick, S. (1982). Local limit theorems for sample extremes, Ann. Prob., 10, 396–413.

    Article  MATH  Google Scholar 

  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York.

    MATH  Google Scholar 

  • Fisher, R.A., and Tippett, L.H.C. (1928). Limiting forms of the frequency distributions of the largest or smallest member of a sample, Proc. Cambridge Philos. Soc., 24, 180–190.

    Article  MATH  Google Scholar 

  • Fréchet, M. (1927). Sur la loi de probabilité de l’écart maximum, Ann. Soc. Polonaise Math. (Cracow), 6, 93.

    Google Scholar 

  • Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed.

    Google Scholar 

  • Krieger, Melbourne, Fla. (1st ed. published in 1978 by Wiley, New York).

    Google Scholar 

  • Geftroy, J. (1958, 1959). Contributions ä la théorie des valeurs extrêmes, Publ. Inst. Statist. Univ. Paris, 7/8, 37–185.

    Google Scholar 

  • Gumbel, E.J. (1958). Statistics of Extremes. Columbia University Press, New York.

    MATH  Google Scholar 

  • Juncosa, M.L. (1949). On the distribution of the minimum in a sequence of mutually independent random variables, Duke Math. J., 16, 609–618.

    Article  MathSciNet  MATH  Google Scholar 

  • Kimber, A.C. (1983). A note on Poisson maxima, Z. Wahrsch. v. Geb., 63, 551–552.

    Article  MathSciNet  MATH  Google Scholar 

  • Leadbetter, M.R (1983). Extremes and local dependence in stationary sequences, Z. Wahrsch. v. Geb., 65, 291–306.

    Article  MathSciNet  MATH  Google Scholar 

  • Leadbetter, M.R., Lindgren, G., and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Series. Springer-Verlag, New York.

    Book  Google Scholar 

  • Leadbetter, M.R., and Rootzén, H. (1988). Extremal theory for stochastic processes, Ann. Prob., 16, 431–478.

    Article  MATH  Google Scholar 

  • Loynes, R.M. (1965). Extreme values in uniformly mixing stationary stochastic processes, Ann. Math. Statist., 36, 993–999.

    Article  MathSciNet  MATH  Google Scholar 

  • Marcus, M.B., and Pinsky, M. (1969). On the domain of attraction of exp(-e-x), J. Math. Anal Appl, 28, 440–449.

    Article  MathSciNet  MATH  Google Scholar 

  • Mejzler, D.G. (1949). On a theorem of B.V. Gnedenko, Sb. Trudov Inst. Mat. Akad. Nauk. Ukrain. SSR, 12, 31–35 (in Russian).

    Google Scholar 

  • Pickands, J. (1968). Moment convergence of sample extremes, Ann. Math. Statist., 39, 881–889.

    Article  MathSciNet  MATH  Google Scholar 

  • Pickands, J. (1975). Statistical inference using extreme order statistics, Ann. Statist., 3, 119–131.

    Article  MathSciNet  MATH  Google Scholar 

  • Pickands, J. (1986). The continuous and differentiable domains of attraction of the extreme value distributions, Ann. Prob., 14, 996–1004.

    Article  MathSciNet  MATH  Google Scholar 

  • Reiss, R.D. (1989). Approximate Distributions of Order Statistics. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Resnick, S. (1987). Extreme Values, Point Processes and Regular Variation. Springer Verlag, New York.

    Google Scholar 

  • Sweeting, T.J. (1985). On domains of uniform local attraction in extreme value theory, Ann. Prob., 13, 196–205.

    Article  MathSciNet  MATH  Google Scholar 

  • von Mises, R. (1936). La distribution de la plus grande de n valeurs, reprinted in Selected Papers II. American Mathematical Society, Providence, R.I. (1954), pp. 271–294.

    Google Scholar 

  • Watson, G.S. (1954). Extreme values in samples from m-dependent stationary stochastic processes, Ann. Math. Statist., 25, 798–800.

    Article  MATH  Google Scholar 

  • Weissman, I. (1975). On location and scale functions of a class of limiting processes with application to extreme value theory, Ann. Prob., 3, 178–181.

    Article  MathSciNet  MATH  Google Scholar 

  • Weissman, I. (1978). Estimation of parameters and large quantites based on the κ largest observations, J. Amer. Statist. Assoc, 73, 812–815.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media New York

About this chapter

Cite this chapter

Smith, R.L. (1992). Introduction to Gnedenko (1943) On the Limiting Distribution of the Maximum Term in a Random Series. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0919-5_13

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0919-5_13

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94037-3

  • Online ISBN: 978-1-4612-0919-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics