Skip to main content

Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions

Abstract

Let \(\left\{ X_n\right\} \) be a sequence of independent and identically distributed non-negative valued random variables defined over a common probability space and let the common distribution function F be continuous. Suppose that F belongs to the domain of partial attraction of a max-semi Frechet law. Define for any \(r\ge 1\) and \( n\ge r\), \( M_{r,[nt]}\) = the rth highest among \(\left\{ X_1,X_2,\ldots ,X_{[nt]}\right\} \) if \( \frac{r}{n} \le t \le 1,\) and = min \(\left\{ X_1,X_2,\ldots ,X_r\right\} \) if \(0\le t \le \frac{r}{n}.\) We establish below, a functional law of the iterated logarithm for the sequence \((M_{r,[nt]}, 0\le t \le 1)\), properly normalized, under the \(M_1\)-topology.

This is a preview of subscription content, access via your institution.

References

  • Breiman L (1987) Probability, Addison-Wesley, Reading

  • Drasin D, Seneta E (1986) A generalization of slowly varying functions. Proc Am Math Soc 96:470–472

    MathSciNet  Article  Google Scholar 

  • de Haan H (1972) The rate of growth of sample maxima. Ann Math Stat 43:1185–1196

    MathSciNet  Article  Google Scholar 

  • Foss S, Korshunov D, Zachary S (2011) An introduction to heavy tailed and sub-exponential distributions. Springer, New york

    Book  Google Scholar 

  • Grinevich IV (1993) Domains of attraction of max-semistable laws under linear and power normalizations. Teor Veroyatn Primen 38:789–799

    MathSciNet  MATH  Google Scholar 

  • Grinevich IV (1994) Max-semi stable laws under linear and power normalizations. Stability problems for Stochastic models ( V.M.Zolotarev et al.(eds.)), 61–70

  • Hall P (1979) On the relative stability of large order statistics. Math Proc Camb Philos Soc 86:467–475

    MathSciNet  Article  Google Scholar 

  • Kiefer J (1971) Iterated logarithm analogues for sample quantiles when p tends to 0. Proc Sixth Berkeley Symp Math Stat Probab 1:227–244

    Google Scholar 

  • Kruglov VM (1972) On the extension of the class of stable distributions. Theor Probab Appl 17:685–694

    MathSciNet  Article  Google Scholar 

  • Mohan NR, Vasudeva R, Hebbar HV (1993) On geometrically infinitely divisible laws and geometric domains of attraction. Sankhya Indian J Stat 55(Ser. A):171–179

    MathSciNet  MATH  Google Scholar 

  • Pancheva E (1995) Max-semistable laws. J Math Sci 76:2177–2180

    MathSciNet  Article  Google Scholar 

  • Seneta E (1976) Regularly varying functions, Lecture notes in Mathematics, 508. Springer, Berlin

  • Skorohod AV (1956) Limit theorems for Stochastic processes. Theor Probab Appl 1:261–290

    MathSciNet  Article  Google Scholar 

  • Vasudeva R, Alireza YM (2011) Kiefer’s law of the iterated logarithm for the vector of order statistics. Probab Math Stat 31:331–347

  • Vasudeva R, Srilakshminarayana G (2013) On strong large deviation results for lightly trimmed sums and some applications. Stat. Probab. Lett. 83:1745–1753

    MathSciNet  Article  Google Scholar 

  • Vasudeva R, Srilakshminarayana G (2015) Almost sure limit points of lightly trimmed sums, when the distribution function belongs to the domain of partial attraction of a positive semi-stable law. Probab. Math. Stat. 35:129–142

    MathSciNet  MATH  Google Scholar 

  • Wichura MJ (1974a) On the functional law of the iterated logarithm for partial maxima of independent and identically distributed random variables. Ann. Probab. 2:202–230

    MathSciNet  MATH  Google Scholar 

  • Wichura MJ (1974b) Functional law of the iterated logarithm for partial sums of iid random variables in the domain of attraction of a completely asymmetric stable law Ann. Probab. 2:1108–1138

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is carried out as a part of the project under the emeritus fellowship of University Grants Commission (UGC), New Delhi, India. The author is grateful to UGC, India, for the support.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Vasudeva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vasudeva, R. Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions. J Indian Soc Probab Stat 22, 357–373 (2021). https://doi.org/10.1007/s41096-021-00108-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41096-021-00108-z

Keywords

  • Max-semi Fréchet law
  • Domain of partial attraction
  • Functional law of the iterated logarithm
  • Almost sure limit function

Mathematics Subject Classification

  • 60F15
  • 60F17
  • 60G17